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2.4: Infinite Limits - Mathematics


Infinite Limits

Evaluating the limit of a function at a point or evaluating the limit of a function from the right and left at a point helps us to characterize the behavior of a function around a given value. As we shall see, we can also describe the behavior of functions that do not have finite limits.

We now turn our attention to (h(x)=1/(x−2)^2), the third and final function introduced at the beginning of this section (see Figure(c)). From its graph we see that as the values of x approach 2, the values of (h(x)=1/(x−2)^2) become larger and larger and, in fact, become infinite. Mathematically, we say that the limit of (h(x)) as x approaches 2 is positive infinity. Symbolically, we express this idea as

[lim_{x o 2}h(x)=+∞.]

More generally, we define infinite limits as follows:

Definitions: infinite limits

We define three types of infinite limits.

Infinite limits from the left: Let (f(x)) be a function defined at all values in an open interval of the form ((b,a)).

i. If the values of (f(x)) increase without bound as the values of x (where (x

ii. If the values of (f(x)) decrease without bound as the values of x (where (x

Infinite limits from the right: Let (f(x)) be a function defined at all values in an open interval of the form ((a,c)).

i. If the values of (f(x)) increase without bound as the values of x (where (x>a)) approach the number (a), then we say that the limit as x approaches a from the left is positive infinity and we write [lim_{x o a+}f(x)=+∞.]

ii. If the values of (f(x)) decrease without bound as the values of x (where (x>a)) approach the number (a), then we say that the limit as x approaches a from the left is negative infinity and we write [lim_{x o a+}f(x)=−∞.]

Two-sided infinite limit: Let (f(x)) be defined for all (x≠a) in an open interval containing (a)

i. If the values of (f(x)) increase without bound as the values of x (where (x≠a)) approach the number (a), then we say that the limit as x approaches a is positive infinity and we write [lim_{x o a} f(x)=+∞.]

ii. If the values of (f(x)) decrease without bound as the values of x (where (x≠a)) approach the number (a), then we say that the limit as x approaches a is negative infinity and we write [lim_{x o a}f(x)=−∞.]

It is important to understand that when we write statements such as (displaystyle lim_{x o a}f(x)=+∞) or (displaystyle lim_{x o a}f(x)=−∞) we are describing the behavior of the function, as we have just defined it. We are not asserting that a limit exists. For the limit of a function f(x) to exist at a, it must approach a real number L as x approaches a. That said, if, for example, (displaystyle lim_{x o a}f(x)=+∞), we always write (displaystyle lim_{x o a}f(x)=+∞) rather than (displaystyle lim_{x o a}f(x)) DNE.

Example (PageIndex{5}): Recognizing an Infinite Limit

Evaluate each of the following limits, if possible. Use a table of functional values and graph (f(x)=1/x) to confirm your conclusion.

  1. (displaystyle lim_{x o 0−} frac{1}{x})
  2. (displaystyle lim_{x o 0+} frac{1}{x})
  3. ( displaystyle lim_{x o 0}frac{1}{x})

Solution

Begin by constructing a table of functional values.

(x)(frac{1}{x})(x)(frac{1}{x})
-0.1-100.110
-0.01-1000.01100
-0.001-10000.0011000
-0.0001-10,0000.000110,000
-0.00001-100,0000.00001100,000
-0.000001-1,000,0000.0000011,000,000

a. The values of (1/x) decrease without bound as (x) approaches 0 from the left. We conclude that

[lim_{x o 0−}frac{1}{x}=−∞. onumber]

b. The values of (1/x) increase without bound as (x) approaches 0 from the right. We conclude that

[lim_{x o 0+}frac{1}{x}=+∞. onumber]

c. Since (displaystyle lim_{x o 0−}frac{1}{x}=−∞) and (displaystyle lim_{x o 0+}frac{1}{x}=+∞) have different values, we conclude that

[lim_{x o 0}frac{1}{x}DNE. onumber]

The graph of (f(x)=1/x) in Figure (PageIndex{8}) confirms these conclusions.

Figure (PageIndex{8}): The graph of (f(x)=1/x) confirms that the limit as x approaches 0 does not exist.

Exercise (PageIndex{5})

Evaluate each of the following limits, if possible. Use a table of functional values and graph (f(x)=1/x^2) to confirm your conclusion.

  1. (displaystyle lim_{x o 0−}frac{1}{x^2})
  2. (displaystyle lim_{x o 0+}frac{1}{x^2})
  3. (displaystyle lim_{x o 0}frac{1}{x^2})

Obviously it depends on the definition of "exists". Some authors explicitly work over the extended real line with $pminfty$ adjoined, so that such infinite limits do explicitly "exist" as first-class values. But there is no consensus. One needs to pay attention to the author's definitions and conventions.

Perhaps it is worth mention - even though this case is rather trivial - that adjoining points at infinity is a special case of various constructions that attempt to simplify matters by some type of existential closure. Below I append an excerpt from my Oct 15, 1996 sci.math post.

This thread originated in a query as to whether infinity or $1/0$ could be admitted as a "value", and soon drifted into discussion of the Riemann sphere and other topological manifestations of infinity via compactification. Below I point out a couple of marvelous references on these topics further I would like to bring to your attention a much wider perspective on such topics, namely that of existential closure as studied in model theory.

There is a beautiful exposition of points at infinity, projective closure, compactifications, modifications, etc. in [FM][1] Chapter 7, Points at Infinity, by H. Behnke and H. Grauert. This is volume III in the excellent "Fundamentals of Mathematics" series, which deserves to be on the bookshelf of every budding mathematician.

A much deeper appreciation of the methodology behind these constructions can be had by studying them from a model-theoretic perspective, in particular from the standpoint of existential closure and model completion. Kenneth Manders has written a series of thought provoking papers [2],[3] from this perspective. I close with an excerpt from the introduction to [2]:

"The systematic adjunction of roots, or solutions to other simple conditions, as in formation of the complex numbers by adjoining imaginaries, or in adjunction of points "at infinity" in traditional geometry, may be analysed as existential closure and model completion. 'Existential closure' refers to a class of processes which attempt to round off a domain and simplify its theory by adjoining elements -- more properly, it refers to the formal relationship that obtains in such a process. 'Model completion' is one of the terms employed when this process is successful. The formation of the complex numbers, and the move from affine to projective geometry, are successes of this kind. Thus, the theory of existential closure gives a theoretical basis of Hilbert's "method of ideal elements." I now sketch the theory of existential closure, to bring out when, how, and in what sense existential closure gives conceptual simplification."

[FM] Fundamentals of mathematics. Vol. III. Analysis.
Edited by H. Behnke, F. Bachmann, K. Fladt and W. Suss.
Translated from the second German edition by S. H. Gould.
Reprint of the 1974 edition. MIT Press,
Cambridge, Mass.-London, 1983. xiii+541 pp. ISBN: 0-262-52095-8 00A05


False Proof – 2 = 4, As the Limit of an Infinite Power Tower

Solution: Consider the value of the following infinitely iterated exponent:

Let , that is, the above power tower where we stop at the -th term. Then is clearly an increasing sequence, and moreover by a trivial induction argument: and if then .

So by basic results in analysis, the sequence of converges to a limit. Let’s call this limit , and try to solve for it.

It is not hard to see that is defined by the equation:

simply because the infinite power tower starting at the second level is the same as the infinite power tower starting at the first level. However, we notice that and both satisfy this relationship. Hence,

Explanation: The issue here is relatively subtle, but one trained in real analysis should have no trouble spotting the problem.

The first alarm is that we are claiming that the limit of a sequence is not unique, which is always false. Inspecting this argument more closely, we corner the flaw to our claim that is defined by the equation . It does, in fact, satisfy this equation, but there is a world of difference between satisfying an equation and being defined by satisfying an equation. One might say that this only makes sense when there is a unique solution to the equation. But as we just saw, there are multiple valid solutions here: 2 and 4 (I don’t immediately see any others it’s clear no other powers of two will work).

Analogously, one might erroneously say that is defined by satisfying the equation , but this does not imply , since the equation has two differing complex roots. This is a slightly murky analogy, as complex conjugation gives an automorphism of . One would reasonably argue that and “play the same role” in this field, while 2 and 4 play wildly different roles as real numbers. But nevertheless, they are not equal simply by virtue of satisfying the same equation.

Moreover, while we said the sequence was bounded from above by 4, an identical argument shows it’s bounded from above by 2. And so one can immediately conclude that the limit of the sequence cannot be 4.

In fact, this infinite tower power is continuous as a function, whose domain is the interval and whose range is . For more information about interesting number-theoretic properties of infinite power towers, see this paper.


Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences

Subsection 3.5.1 Limits at Infinity

We occasionally want to know what happens to some quantity when a variable gets very large or “goes to infinity”.

Example 3.18 . Limit at Infinity.

What happens to the function (ds cos(1/x)) as (x) goes to infinity? It seems clear that as (x) gets larger and larger, (1/x) gets closer and closer to zero, so (cos(1/x)) should be getting closer and closer to (cos(0)=1 ext<.>)

As with ordinary limits, this concept of “limit at infinity” can be made precise. Roughly, we want (ds lim_f(x)=L) to mean that we can make (f(x)) as close as we want to (L) by making (x) large enough.

Definition 3.19 . Limit at Infinity.

if (f(x)) can be made arbitrarily close to (L) by taking (x) large enough. If this limits exists, we say that the function (f) has the limit (L) as (x) increases without bound.

if (f(x)) can be made arbitrarily close to (M) by taking (x) to be negative and sufficiently large in absolute value. If this limit exists, we say that the function (f) has the limit (L) as (x) decreases without bound.

Example 3.20 . Limit at Infinity.

Let (f) and (g) be the functions

Referring to the graphs of (f(x)) and (g(x)) shown below, we see that

Definition 3.21 . Limit at Infinity (Formal Definition).

If (f) is a function, we say that (ds lim_f(x)=L) if for every (epsilon>0) there is an (N > 0) so that whenever (x>N ext<,>) (|f(x)-L|lt epsilon ext<.>) We may similarly define (ds lim_f(x)=L ext<.>)

We include this definition for completeness, but we will not explore it in detail. Suffice it to say that such limits behave in much the same way that ordinary limits do in particular there is a direct analog of Theorem 3.9.

Example 3.22 . Limit at Infinity.

As (x) goes to infinity both the numerator and denominator go to infinity. We divide the numerator and denominator by (ds x^2 ext<:>)

Now as (x) approaches infinity, all the quotients with some power of (x) in the denominator approach zero, leaving 2 in the numerator and 1 in the denominator, so the limit again is 2,

In the previous example, we divided by the highest power of (x) that occurs in the denominator in order to evaluate the limit. We illustrate another technique similar to this.

Example 3.23 . More Limits at Infinity.

Compute the following limit:

As (x) becomes large, both the numerator and denominator become large, so it isn't clear what happens to their ratio. The highest power of (x) in the denominator is (x^2 ext<,>) therefore we will divide every term in both the numerator and denominator by (x^2) as follows:

We can also apply limit laws to infinite limits instead of arguing as we did in Example 3.20.

Note that we used the theorem above to get that (dsfrac<1>=0>) and (dsfrac<1>=0> ext<.>)

A shortcut technique is to analyze only the leading terms of the numerator and denominator. A leading term is a term that has the highest power of (x ext<.>) If there are multiple terms with the same exponent, you must include all of them.

Top: The leading term is (2x^2 ext<.>)

Bottom: The leading term is (5x^2 ext<.>)

Now only looking at leading terms and ignoring the other terms we get:

Example 3.24 . Application of Limits at Infinity.

A certain manufacturer makes a line of luxurious chairs. It is estimated that the total cost of making (x) luxurious chairs is

dollars per year. Thus, the average cost of making (x) chairs is given by

dollars per chair. Evaluate (limlimits_ overline) and interpret your results.

A sketch of the graph of the function (overline) is shown below. The result we obtained is fully expected if we consider its economic implications. Note that as the level of production increases, the fixed cost per chair produced, represented by the term (dfrac<200,000> ext<,>) drops steadily. The average cost should approach a constant unit cost of production — $350 in this case (see below).

Use the slider below to investigate the limit (dslim_ overline ext<,>) where (overline = 350 + dfrac<200,000> ) as found in Example 3.24.

Subsection 3.5.2 Infinite Limits

We next look at functions whose limit at (x = a) does not exist, but whose values increase or decrease without bound as (x) approaches (a) from the left or right.

Definition 3.25 . Infinite Limit (Useable Definition).

if we can make the value of (f(x)) arbitrarily large by taking (x) to be sufficiently close to (a) (on either side of (a)) but not equal to (a ext<.>) Similarly, we write

if we can make the value of (f(x)) arbitrarily large and lue by taking (x) to be sufficiently close to (a) (on either side of (a)) but not equal to (a ext<.>)

We want to emphasize that by the proper definition of limits, the above limits do not exist, since they are not real numbers. However, writing (pm infty) provides us with more information than simply writing DNE.

This definition can be modified for one-sided limits as well as limits with (x o a) replaced by (x oinfty) or (x o-infty ext<.>)

Example 3.26 . Simple Infinite Limit.

Compute the following limit: (dslimlimits_ frac<1> ext<.>)

We refer to the graph below. Let's first look at the limit as (x o 0^+ ext<,>) and notice that (frac<1>) increases without bound. Therefore,

As (x o 0^- ext<,>) we again see that (frac<1>) increases without bound:

Example 3.27 . Infinite Limit.

Compute the following limit: (dslim_(x^3-x) ext<.>)

One might be tempted to write:

however, we do not know what (infty-infty) is, as (infty) is not a real number and so cannot be treated like one. Incidentally, the expression (infty - infty) is another indeterminate form.

As (x) becomes arbitrarily large, then both (x) and (x^2-1) become arbitrarily large, and hence their product (x(x^2-1)) will also become arbitrarily large. Thus we see that

Example 3.28 . More Infinite Limit.

Dividing the numerator and the denominator of the rational expression by (x^<2> ext<,>) we obtain

Since the numerator becomes arbitrarily large whereas the denominator approaches (1) as (x) tends to infinity, we see that the quotient (f(x)) gets larger and larger as (x) approaches infinity. In other words, the limit does not exist. We indicate this by writing

Once again, dividing both the numerator and the denominator by (x^<2> ext<,>) we obtain

In this case, the numerator becomes arbitrarily large in magnitude, bu negative in sign, whereas the denominator approaches (1) as (x) appraoches negative infinity. Therefore, the quotient (f(x)) decreases without bound, and the limit does not exists. We indicate this by writing

Example 3.29 . Limit at Infinity, Infinite Limit and Basic Functions.

Find the following limits by observing the behaviour of the graph of each function.

We can easily evaluate the following limits by observation:

Making use of the results from Example 3.29 we can compute the following limits.

Example 3.30 . More Limit at Infinity, Infinite Limit and Basic Functions.

Thus, as left-hand limit ( eq) right-hand limit,

Subsection 3.5.3 Vertical Asymptotes

Definition 3.31 . Vertical Asymptote.

The line (x=a) is called a of (f(x)) if of the following is true:

Use the slider below to investigate the limit (ds lim_> f(x) ext<.>)

Example 3.32 . Vertical Asymptotes.

Find the vertical asymptotes of (ds f(x)=frac<2x> ext<.>)

In the definition of vertical asymptotes we need a certain limit to be (pminfty ext<.>) Candidates would be to consider values not in the domain of (f(x) ext<,>) such as (a=4 ext<.>) As (x) approaches (4) but is larger than (4) then (x-4) is a small positive number and (2x) is close to (8 ext<,>) so the quotient (2x/(x-4)) is a large positive number. Thus we see that

Thus, at least one of the conditions in the definition above is satisfied. Therefore (x=4) is a vertical asymptote, as shown below.

Subsection 3.5.4 Horizontal Asymptotes

Definition 3.33 . Horizontal Asymptote.

The line (y=L) is a of (f(x)) if either

Example 3.34 . Horizontal Asymptotes.

Find the horizontal asymptotes of (ds f(x)=frac<|x|> ext<.>)

We must compute two infinite limits. First,

Notice that for (x) arbitrarily large that (x>0 ext<,>) so that (|x|=x ext<.>) In particular, for (x) in the interval ((0,infty)) we have

Notice that for (x) arbitrarily large negative that (xlt 0 ext<,>) so that (|x|=-x ext<.>) In particular, for (x) in the interval ((-infty,0)) we have

Therefore there are two horizontal asymptotes, namely, (y=1) and (y=-1 ext<,>) as shown to the right.

Subsection 3.5.5 Slant Asymptotes

Some functions may have slant (or oblique) asymptotes, which are neither vertical nor horizontal.

Definition 3.35 . Slant Asymptote.

The line (y=mx+b) is a of (f(x)) if either

Visually, the vertical distance between (f(x)) and (y=mx+b) is decreasing towards 0 and the curves do not intersect or cross at any point as (x) approaches infinity.

Example 3.36 . Slant Asymptote in a Rational Function.

Find the slant asymptotes of (ds f(x)=frac<-3x^2+4> ext<.>)

Note that this function has no horizontal asymptotes since (f(x) o-infty) as (x oinfty) and (f(x) oinfty) as (x o-infty ext<.>)

In rational functions, slant asymptotes occur when the degree in the numerator is one greater than in the denominator. We use long division to rearrange the function:

The part we're interested in is the resulting polynomial (-3x-3 ext<.>) This is the line (y=mx+b) we were seeking, where (m=-3) and (b=-3 ext<.>) Notice that

Thus, (y=-3x-3) is a slant asymptote of (f(x) ext<,>) as shown in Figure 3.1 below.

Although rational functions are the most common type of function we encounter with slant asymptotes, there are other types of functions we can consider that present an interesting challenge.

Example 3.37 . Slant Asymptote.

Show that (y=2x+4) is a slant asymptote of (f(x)=2x-3^x+4 ext<.>)

We note that (lim_[f(x)-(2x+4)]=lim_(-3^x)=-infty ext<.>) So, the vertical distance between (y=f(x)) and the line (y=2x+4) decreases toward 0 only when (x o -infty) and not when (x o infty ext<.>) The graph of (f) approaches the slant asymptote (y=2x+4) only at the far left and not at the far right. One might ask if (y=f(x)) approaches a slant asymptote when (x o infty ext<.>) The answer turns out to be no, but we will need to know something about the relative growth rates of the exponential functions and linear functions in order to prove this. Specifically, one can prove that when the base is greater than 1 the exponential functions grows faster than any power function as (x o infty ext<.>) This can be phrased like this: For any (a>1) and any (n>0 ext<,>)

These facts are most easily proved with the aim of something called the L'Hôpital's Rule.

Subsection 3.5.6 End Behaviour and Comparative Growth Rates

Let us now look at the last two subsections and go deeper. In the last two subsections we looked at horizontal and slant asymptotes. Both are special cases of the end behaviour of functions, and both concern situations where the graph of a function approaches a straight line as (x o infty) or (-infty ext<.>) But not all functions have this kind of end behaviour. For example, (f(x)=x^2) and (f(x)=x^3) do not approach a straight line as (x o infty) or (-infty ext<.>) The best we can say with the notion of limit developed at this stage are that

Similarly, we can describe the end behaviour of transcendental functions such as (f(x)=e^x) using limits, and in this case, the graph approaches a line as (x o -infty) but not as (x o infty ext<.>)

People have found it useful to make a finer distinction between these end behaviours all thus far captured by the symbols (infty) and (-infty ext<.>) Specifically, we will see that the above functions have different growth rates at infinity. Some increases to infinity faster than others. Specifically,

Definition 3.38 . Comparative Growth Rates.

Suppose that (f) and (g) are two functions such that (limlimits_fleft( x ight) =infty) and (limlimits_gleft( x ight) =infty ext<.>) We say that (fleft( x ight)) grows faster than (gleft( x ight)) as (x o infty) if the following holds:

Here are a few obvious examples:

Example 3.39 . Growth Rate in Polynomial Functions.

Show that if (m>n) are two positive integers, then (f(x)=x^m) grows faster than (g(x)=x^n) as (x o infty ext<.>)

Since (m>n ext<,>) (m-n) is a positive integer. Therefore,

Example 3.40 . Growth Rate in Polynomial Functions.

Show that if (m>n) are two positive integers, then any monic polynomial (P_(x)) of degree (m) grows faster than any monic polynomial (P_(x)) of degree (n) as (x o infty ext<.>) [Recall that a polynomial is monic if its leading coefficient is 1.]

By assumption, (P_(x)=x^m+) terms of degrees less than (m=x^+a_x^+ldots ext<,>) and (P_(x)=x^n+) terms of degrees less than (n=x^+b_x^+ldots ext<.>) Dividing the numerator and denominator by (x^n ext<,>) we get

since the limit of the bracketed fraction is 1 and the limit of (x^) is (infty ext<,>) as we showed in Example 3.39.

Example 3.41 . Growth Rate in Polynomial Functions.

Show that a polynomial grows exactly as fast as its highest degree term as (x o infty) or (-infty ext<.>) That is, if (P(x)) is any polynomial and (Q(x)) is its highest degree term, then both limits

Suppose that (P(x)=a_x^+a_x^+ldots+a_<1>x+a_<0> ext<,>) where (a_ eq 0 ext<.>) Then the highest degree term is (Qleft( x ight) =a_x^ ext<.>) So,

Let's state a theorem we mentioned when we discussed the last example in the last subsection:

Theorem 3.42 . Growth Comparison Between Exponential and Power Functions.

Let (a) and (n) be positive real numbers with (a>1 ext<.>) Then (f(x)=a^x) grows faster than (g(x)=x^n) as (x o infty ext<:>)

The easiest way to prove this is to use the L'Hôpital's Rule, which we will introduce in a later chapter. For now, one can plot and compare the graphs of an exponential function and a power function. Here is a comparison between (f(x)=x^2) and (g(x)=2^x ext<:>)

Notice also that as (x o -infty ext<,>) (x^n) grows in size but (e^x) does not. More specifically, (x^n o infty) or (-infty) according as (n) is even or odd, while (e^x o 0 ext<.>) So, it is meaningless to compare their “growth” rates, although we can still calculate the limit

Let's see an application of our theorem.

Example 3.43 . Horizontal Asymptotes.

Find the horizontal asymptote(s) of (f(x)=dfrac ext<.>)

To find horizontal asymptotes, we calculate the limits of (f(x)) as (x o infty) and (x o -infty ext<.>) For (x o infty ext<,>) we divide the numerator and the denominator by (e^x ext<,>) and then we take limit to get

For (x o -infty ext<,>) we divide the numerator and the denominator by (x^2) to get

The denominator now approaches (0-4=-4 ext<.>) The numerator has limit (-infty ext<.>) So, the quotient has limit (infty ext<:>)

So, (y=2) is a horizontal asymptote. The function (y=f(x)) approaches the line (y=2) as (x o infty ext<.>) And this is the only horizontal asymptote, since the function (y=f(x)) does not approach any horizontal line as (x o -infty ext<.>)

Since the growth rate of a polynomial is the same as that of its leading term, the following is obvious:

Example 3.44 . Growth Rate in Polynomial and Exponential Functions.

If (P(x)) is any polynomial, then

Also, if (r) is any real number, then we can place it between two consecutive integers (n) and (n+1 ext<.>) For example, (sqrt<3>) is between 1 and 2, (e) is between 2 and 3, and (pi) is between 3 and 4. Then the following is totally within our expectation:

Example 3.45 . Growth Rate in Exponential Functions.

Prove that if (a>1) is any basis and (r>0) is any exponent, then (f(x)=a^x) grows faster than (g(x)=x^r) as (x o infty ext<.>)

Let (r) be between consecutive integers (n) and (n+1 ext<.>) Then for all (x>1 ext<,>) (x^leq x^leq x^ ext<.>) Dividing by (a^ ext<,>) we get

What about exponential functions with different bases? We recall from the graphs of the exponential functions that for any base (a>1 ext<,>)

So, the exponential functions with bases greater than 1 all grow to infinity as (x o infty ext<.>) How do their growth rates compare?

Theorem 3.46 . More Growth Comparison between Exponential Functions.

If (1lt alt b ext<,>) then (f(x)=b^x) grows faster than (g(x)=a^x) as (x o infty ext<:>)

Proof.

Another function that grows to infinity as (x o infty) is (g(x)=ln x ext<.>) Recall that the natural logarithmic function is the inverse of the exponential function (y=e^x ext<.>) Since (e^x) grows very fast as (x) increases, we should expect (ln x) to grow very slowly as (x) increases. The same applies to logarithmic functions with any basis (a>1 ext<.>) This is the content of the next theorem.

Theorem 3.47 . Growth Comparison between Logarithmic and Power Functions.

Let (a) and (n) be any positive real numbers with (a>1 ext<.>) Then (f(x)=x^n) grows faster than (g(x)=log_a x) as (x o infty ext<:>)

Proof.

We use a change of variable. Letting (t=ln x ext<,>) then (x=e^t ext<.>) So, (x o infty) if and only if (t o infty ext<,>) and

Now, since (r>0 ext<,>) (a=e^r>1 ext<.>) So, (a^t) grows as (t) increases, and it grows faster than (t) as (t o infty ext<.>) Therefore,


. divide the coefficients of the terms with the largest exponent, like this:

(note that the largest exponents are equal, as the degree is equal)

. then the limit is positive infinity .

. or maybe negative infinity. We need to look at the signs!

We can work out the sign (positive or negative) by looking at the signs of the terms with the largest exponent, just like how we found the coefficients above:

For example this will go to positive infinity, because both .

  • x 3 (the term with the largest exponent in the top) and
  • 6x 2 (the term with the largest exponent in the bottom)

Infinite Limits---The $frac n 0$ Form

Since the limit has the $frac< ed n><0>$ form, we know the limit does not exist. However, it still might be an infinite limit.

Examine the left-hand limit.

  • The numerator is positive.
  • Since the denominator is $x^2$, it will be positive.
  • The $frac< ed n><0>$ form tells us the function is becoming infinitely large.

Taken together, these three statements tell us $displaystyle lim_,frac 1 = infty$

Examine the right-hand limit.

  • The numerator is positive.
  • Since the denominator is $x^2$, it will be positive.
  • The $frac< ed n><0>$ form tells us the function is becoming infinitely large.

Taken together, these three statements tell us $displaystyle lim_,frac 1 = infty$

Both one-sided limits tell us the function is growing infinitely large in the positive direction. Since the one-sided limits indicate the same behavior, the limit is infinite.

Answer: $displaystyle lim_,frac 1 = infty$

Example 2

Evaluate: $displaystyle lim_,frac 1 $

Determine the form of the limit.

The limit does not exist since it has the $frac< ed n><0>$ form. It might also be an infinite limit.

Examine the left-hand limit.

  1. The numerator is always positive.
  2. The denominator will be negative, since $x Step 3

Examine the right-hand limit.

  1. The numerator is always positive.
  2. The denominator will be positive, since $x>3$.
  3. The function is becoming infinitely large.

Taken together, these statements tell us $displaystyle lim_,frac 1 = infty$

Since the one-sided limits are different, the limit does not exist.

Answer: $displaystyle lim_,frac 1 $ does not exist.

Example 3

Determine the form of the limit.

The limit does not exist, but it might be an infinite limit.

Factor the denominator to make our analysis easier.

Examine the one-sided limits.

  1. For both limits, the numerator will be -3.
  2. Since the denominator is being squared, it will always be positive.
  3. The function becomes infinitely large as $x$ approaches 2.

Since the numerator and denominator have opposite signs, the function will grow infinitely large in the negative direction.


Infinite Limits---Skill Practice

The limit does not exist, but it has the necessary form so that it might be an infinite limit.

Examine the left-hand limit.

  1. The numerator approaches 5, so it will be positive.
  2. Since $x$ is approaching 3 from the left, the denominator will be negative.
  3. As the denominator shrinks to 0, the function becomes infinitely large.

Result: $displaystyle lim_,frac = -infty$

Examine the right-hand limit.

  1. The numerator approaches 5, so it will be positive.
  2. Since $x$ is approaching 3 from the right, the denominator will be positive.
  3. As the denominator shrinks to 0, the function becomes infinitely large.

Result: $displaystyle lim_,frac = infty$

$displaystyle lim_,frac$ does not exist.

Example 2

Determine the form of the limit.

The limit doesn't exist, but it has the $frac n 0$ form so it might be an infinite limit.

Try factoring the denominator so the one-sided limits are easier to analyze.

Examine the one-sided limits.

  1. In both cases, the numerator approaches -1, so the numerator will be negative.
  2. In both cases, the denominator is being squared, so it will always be positive.
  3. In both cases, the denominator is approaching 0, so the function will become infinitely large.

Both one-sided limits grow infinitely large in the negative direction.

Example 3

Determine the form of the limit.

Find and divide out any common factors.

We know the limit doesn't exist. Since it has the $frac n 0$ form, it might be an infinite limit.

Examine the left-hand limit.

  1. The numerator approaches 8, so it will be positive.
  2. Since $x$ is approaching 4 from the left, the denominator will be negative.
  3. As the denominator shrinks to 0, the function will become infinitely large.

Examine the right-hand limit.

  1. The numerator approaches 8, so it is positive.
  2. Since $x$ is approaching 4 from the right, the denominator will be positive.
  3. As the denominator shrinks to 0, the function will become infinitely large.
Example 4

Evaluate the limit to determine its form.

The limit doesn't exist, but it has the $frac n 0$ form so it might be an infinite limit.

Examine the left-hand limit.

  1. The numerator approaches 95, so it will be positive.
  2. Since $x$ is approaching 5 from the left, we know $x Step 3


The most important properties of limits are the algebraic properties, which say essentially that limits respect algebraic operations:

These can all be proved via application of the epsilon-delta definition. Note that the results are only true if the limits of the individual functions exist: if lim ⁡ x → a f ( x ) limlimits_ f(x) x → a lim ​ f ( x ) and lim ⁡ x → a g ( x ) limlimits_ g(x) x → a lim ​ g ( x ) do not exist, the limit of their sum (or difference, product, or quotient) might nevertheless exist.

This is an example of continuity, or what is sometimes called limits by substitution.


INFINITY IN THEOLOGY AND MATHEMATICS

Can we apply the same concepts to both the finite and the infinite? Is there something distinctive about the infinite that prevents attribution to it of concepts that we can attribute to the finite? If so, then this could be a reason for our difficulties in talking about God &mdash God is infinite, and our concepts, applying, as they do, to the finite objects of our experience, cannot be `extended' to the infinite. God's infinity is sometimes used as an explanation of theological difficulties like the problem of evil or the paradoxes of omnipotence: we do not really know what we mean when we attribute infinite goodness or power to God.

The mathematical concept of the infinite may be able to shed some light on the theological concept. In mathematics, there is a clearly developed concept of the infinite. We might expect that if there are problems of expressibility related to infinity, they will appear in mathematics as well as in theology. Since the mathematical concept is clearer, the source of any difficulties with it may be clearer as well.

I shall suggest the following conclusions. First, the difficulty that we shall see in pinning down the meaning of `infinite' in its application to God is partly the result of there being two disparate trends in theology: one that tries to find an answer to the problems that come from God's infinity, and one that insists on the impenetrable mystery of the infinite. Second, the development of the mathematical concept of the infinite can shed light on the theological concept not so much by indicating the possibility of a parallel development and clarification of God's infinity but rather by showing that the attribution of infinity to God may well be &mdash at the present stage of development of the mathematical concept &mdash more figurative than literal. The concepts of the mathematical infinite and the theological infinite may at one time have been quite close but by now there is at most a metaphorical connexion. And third, infinity is less a cause or a reason for inexpressibility than an attribution resulting from the experience of God as in some degree mysterious.

1. HISTORICAL SURVEY OF THE INFINITE

The following general classifications of infinity are to be developed and refined through a brief historical survey: first, the potential infinite second, the actual infinite and third, the theological infinite.

Even in mathematics, there is not a simple dichotomy between the finite and the infinite. Opposed to the finite are the potential infinite and the actual infinite. The potential infinite is an extension of the finite, constructible from the finite by some rule or process that is never in fact completed. The actual infinite, on the other hand, is conceived as an actually existing collection of an infinite number of parts. The actual infinite could be regarded as the completion of a process that builds the infinite from the finite. This is the actual infinite of set theory.

God's infinity, however, is not conceived as one of these species of mathematical infinity. Descartes and Leibniz, for example, make a further distinction between the mathematical infinite and what they call the absolute infinite (and which I shall call the theological infinite). The conception of the theological infinite is not a conception of an infinite collection, but rather of the unbounded or unlimited.

1.1 The mathematical infinite

Aristotle distinguished between the potential infinite and the actual infinite. In none of the experiential sources for the concept of the infinite does the infinite actually exist. It exists only potentially, by addition or division. Aristotle argues that saying that the infinite potentially-is, is not like saying that a statute potentially-is. If a statute potentially-is, that means that there will be an actual statute. The infinite does not potentially exist in this sense &mdash there will be no actual infinite. Rather, to say `there is the infinite', according to Aristotle, means that one thing after another will be coming into being. The infinite takes place in a succession of different finite things: by the division of the parts in magnitude or by addition in the succession of moments or beings. The infinite has being only as a process that can be repeated over and over again without end, but which, at any moment, has only a finite number of components. 1 [Aristotle, Physica, translated by R.P. Hardie and R.K. Gaye, in The Basic Works of Aristotle, edited by Richard McKeon (N.Y.: Random House, 1941), 206a. ]

Aristotle's idea that the infinite has being only potentially and not actually was denied by Cantor. In Cantor's view, a set has to be regarded as a whole, as a totality with all the parts simultaneous. This appears to obviate the distinction between the potential and the actual infinite.

Cantor's actual infinite may not be the same as Aristotle's actual infinite, even though Cantor represents it as such. As Bochner points out, 2 [Salomon Bochner, `Infinity', in Dictionary of the History of Ideas, vol. 11 (N.Y.: Charles Scribner's Sons, 1973), p. 612.] Aristotle was not talking about number, but about body &mdash physics rather than mathematics. On the other hand, Cantor's actual infinite is in principle constructible by means of a rule or process. Whereas for Aristotle the infinite has being only as this infinitely repeatable rule or process, Cantor's actual infinite could be seen as the completion of such a process. This is not to suggest that the actual infinite is defined by any unique process of construction, but rather that the actual infinite can be seen as what would result from the endless repetition of a process. In any case, this process requires a `subjunctive leap': the actual infinite is not in fact the result of repeating the process an infinite number of times, but rather represents what would result from infinite repetitions. (Similarly, the mathematical statement of Zeno's paradox converges to 1, but determining this limit does not require actually adding up the infinite sum the limit tells you what result you would get if you had added up the infinite sum.)

Cantor claims that a distinction still remains between kinds of infinite: there are consistent multiplicities, which are infinite sets like the sets of integers and there are inconsistent multiplicities. These are the infinite sets that in one way or another cause a problem. As he says,

a multiplicity can be such that the assumption that all its elements `are together' leads to a contradiction, so that it is impossible to conceive of the multiplicity as a unity, as `one finished thing' . the `totality of everything thinkable', for example, is such a multiplicity later still other examples will turn up. 3 [George Cantor, `Letter to Dedekind', in From Frege to Godel: A Source Book in Mathematical Logic, ed. Jean van Heijenoort (Cambridge, Mass. and London: Harvard University Press, 1967), p. 114.]

It does not seem likely, however, that Cantor's inconsistent sets point to another species of infinity, or to a problem about infinity as such. Cantor's paradox and its cognates do not rely at all upon the size of their domain &mdash they can be stated as well for finite sets. In other words, although there can be inconsistent infinite sets, there can also be finite sets that are inconsistent for the same reason.

1.2 The theological infinite

The actual infinite, as I have said, can be conceived as collection of an infinite number of parts, the completion of some process that builds the infinite from the finite. The only problem with actually infinite sets, if there is a problem with them, is that there is not enough time to build them. Nevertheless, they are still the kinds of things that could be built from a collection of parts. The theological infinite, however, is not conceived as an infinite collection, but rather as the unbounded or unlimited it is not in any sense constructible from the finite because it is not a collection at all, not an extension of the finite.

Descartes puts this as a distinction between the infinite and the indefinite, between a positive and a negative idea. The actual or potential infinite of mathematics is more properly called indefinite only God is infinite. Indefinite things are those in which we observe no limits, and perhaps can conceive no limits but we cannot prove that they must have no limits. What we can call infinite, on the other hand, is that in which we not only observe no limits, but also can be certain that there can be no limits. 4 [Descartes, Principles of Philosophy, in The Philosophical Works of Descartes, 1, trans. Elizabeth S. Haldane and G.R.T. Ross (Cambridge: Cambridge University Press, 1931), 1, xxvii.]

God is the only thing I positively conceive as infinite. As to other things like the extension of the world and the number of parts into which matter is divisible, I confess I do not know whether they are absolutely infinite I merely know that I can see no end to them. . . . 5 [Descartes, letter to More, 5 February 1649, in Philosophical Letters, trans., ed. Anthony Kenny (Minneapolis: University of Minnesota Press, 1981), p. 242.]

We have a positive understanding that God is without limits, but

in regard to other things . we do not in the same way positively understand them to be in every part unlimited, but merely admit that their limits, if they exist, cannot be discovered by us. 6 [Descartes, Principles of Philosophy, 1, xxvii.]

So Descartes says that there are things other than God that are apparently without limits but he separates the unlimitedness of these other things from the unlimitedness of God by saying that only in the case of God are we certain that there can be no limits. In all other cases, according to Descartes, we can say only that we know of no limits but there may or may not still be limits.

The idea of the theological infinite is, for Descartes, a different idea from that of the mathematical potential or actual infinite. There is no way that the idea of the infinite can be derived from the indefinite &mdash the idea of the infinite is a positive idea that cannot be constructed from the negative idea of the indefinite. This distinction between the indefinite and the infinite has a crucial part in Descartes' proof for the existence of God: since the idea of God is the idea of an actually infinite being, and I am at most potentially infinite, I could not be responsible for this idea myself. 7 [Descartes, Meditations, in Haldane and Ross, pp. 166f.]

This distinction of the infinite from the merely indefinite may be apt for some objects &mdash we cannot at present prove, for instance, that there are or that there are not ultimate constituents that would limit divisibility of matter. But what about number? We do seem to be able to prove that number is infinite, in virtue of the fact, for example, that the series of positive integers can be put into a one-one correspondence with a proper subset of itself. This is a positive proof of infinity, of the sort that Descartes would reserve only for God. In separating God's infinity from the merely indefinite of mathematics, Descartes perhaps means to retain God's infinity as something absolutely unique, mysterious and strange, in the face of which we stand in awe. But Descartes' definitions of the infinite and the indefinite do not provide as much support for this strangeness as Descartes intended.

In Leibniz there may be somewhat stronger support for the idea that God's infinity is a different sort of thing from mathematical infinity. Leibniz rejects the actual infinite, in the sense of an actually existing whole made up of an infinite number of parts: such a conception, he says, is of use only in mathematics. The infinite analysis of concepts can help to show that God's infinity is a different sort of thing. This infinite analysis is for us a potential infinite. Similarly, a line has an infinite number of points, but these points are present for us only potentially. As this infinite number of points has a use in mathematics, so the infinite analysis of a concept is of use in metaphysics: neither infinite is actual in the sense that it can be grasped by us simultaneously in all its parts. God, on the other hand, is able to grasp the complete concept of a thing. God does not, however, get this understanding by completing the infinite analysis of the concept instead, he grasps the whole thing all at once. God is not, as you might say, isomorphic with the infinite series, such that his nature is peculiarly suited to carrying out an infinite analysis. Rather, this understanding is not the result of a serial analysis, not the result of carrying to completion the process.

Leibniz, like Descartes, makes God's infinity something other than another species of mathematical infinity. In a sense our idea of the infinite is for Leibniz derived from the finite and from the impossibility that we should ever come to the end of our ability to continue adding or dividing. But our ability thus to build the infinite from the finite, to understand that we can keep applying the same rules or processes over and over, getting a new result each time, is grounded in the idea of the theological infinite. The idea of the theological infinite, which grounds the mathematical infinite, is an idea of an attribute of God. (That is, God is infinite and God's other qualities are in God in the manner proper to an infinite being. Qualities are determined and limited by the nature of the being in which they inhere in an infinite being, there is no limitation of the quality by the nature of the being. 8 [This explication of God's infinity is close to Aquinas's: cf. Summa Theologica 17.] ) This absolute infinite `precedes all composition and is not formed by the addition of parts'. It is not, that is, a completed whole, but rather, `an attribute with no limits'. 9 [Leibniz, New Essays on Human Understanding, trans. Peter Remnant and Jonathan Bennett (Cambridge: Cambridge University Press, 1981), 157ff.]

II. POSSIBLE DEFINITIONS OF THE UNLIMITED

The theological infinite, then, is distinguished from the mathematical infinite in that the theological infinite is defined only by the lack of limits or bounds. This could define the mathematical infinite in only the most naive way &mdash we could say, for instance, that there is no end to number. We have to consider what a limit is, in this sense, and what it means to say that something has or does not have limits. Mathematics shows that the infinite cannot be simply identified with the unlimited: there can be a space that is unlimited but finite and the real numbers between 0 and 1 are an example of the limited but infinite.

It seems that the idea of the unlimited incorporates these factors: the idea that goodness, power and other qualities are not limited by the nature of God as they are limited by the nature of the other beings in which they are found the idea that the quality is inexhaustible in God and the idea that God is a sort of `upper bound' of the quality. But none of these conclusively defines `unlimited' on its own nor do they add up to a conclusive definition.

2.1 Completeness

In his historical survey of the concept of the infinite, S. Bochner has suggested that the theological infinite is a qualitative, rather than a quantitative, infinite. The idea of a qualitative infinite is not at all clear, however. The qualitatively infinite might be seen as something that `expresses a degree of completeness or perfection of something structurable'. 10 [Bochner, `Infinity', p. 614.] According to Aristotle, completeness rules out infinity altogether: what is complete has nothing lacking to it, and thus nothing outside of it but the infinite always has something outside it &mdash `outside' in the sense that, since the infinite is an unending process, there is always something that this process has not yet incorporated: another unit to be added or another division to be made. 11 [Aristotle, Physica, 206b-207a.] Bochner points out that mathematizations of completeness show that completeness need not rule out infinity but to say that completeness need not rule out infinity is obviously not to say that infinity can be defined in terms of completeness. It might be suggested that in the qualitatively infinite the quality is infinitely present. This does not, however, seem to add anything to the idea of the unlimited. To say that in God being, goodness, power and so forth are infinitely present is to say no more than that the quality is not limited by the nature of God, as it is limited by the nature of any creature in which it is found.

2.2 Inexhaustibility

The notion of inexhaustibility, of not being able ever to come to the end, is also relevant here. For instance, you could never come to the end of God's infinite mercy: you could never sin to much or too often to be forgiven. Inexhaustibility cannot define the mathematical infinite &mdash something could be practically or virtually inexhaustible, and yet be (hugely) finite. The class of the first 10 100 positive integers is inexhaustibly large in the sense that you could never come to the end of counting them but it is not infinite. On the other hand, we have seen that the mathematical and the theological conceptions of the infinite are not the same. So the fact that inexhaustibility does not define the mathematical infinite is not conclusive evidence against its defining the theological infinite.

2.3 Maximality and purity

We might try defining theological infinity as maximality &mdash so that saying that God is infinitely good, for instance, means that God is more good than anything else possible. This `upper bound' conception of God's infinity is close to the definition of God as `that than which there can be no greater'. The fact that this definition is relative to a class of possible things, rather than to a class of actual things, means that at least there is nothing contingent or variable about the definition.

This definition of the theological infinite, however, does not fully answer the question of what we mean when we attribute infinity to God. The impact of the infinite on our ability to think about and talk about God is determined by a decision to accept or reject what we might call a `purity postulate'. By `purity' I mean the notion that if God is infinitely good, then there is in God no trace or mixture of evil if God is powerful, there is in God no trace or mixture of inability and so forth. We can accept or reject this purity postulate while retaining the definition of God's infinity as maximality. It seems that this postulate of purity is independent of the attribution of infinity to God.

Clearly something could be more good than anything else possible, and still not be purely good, more powerful than anything else possible, and still not purely powerful, and so forth. And in this case, God could still be said to be infinitely powerful &mdash to be more powerful than anything else possible &mdash but still not be powerful enough to eradicate evil altogether. In this case, pure power, pure goodness, would be an impossibility. On the other hand, we might say that this does not capture adequately what we mean to say when we say that God is unlimited. We might want to say that if God's power is infinite, then there cannot be a certain point beyond which it does not extend. If there were, that would be a limit and hence God's power would be only hugely finite.

The point is that there is no agreement about how far God's lack of limits reaches, no agreement about the acceptance or rejection of this idea of purity. Both Descartes and Leibniz define theological infinity as the lack of limits. But Leibniz does not claim that God can contravene the laws of logic and Descartes does. For Leibniz, God's power is unlimited in the sense that God can do anything logically possible. For Descartes, God's power is unlimited altogether. Descartes says that God could make a mountain without a valley, or make it the case that 2 + 3 5: that we cannot conceive these things is no indication of a limit to God's power. We need not see a paradox or a contradiction as indicating a limit to God. We can see it, instead, as indicating a limit to our minds.

2.4 Mystery

The idea of God's infinite goodness as the upper bound of possible goodness brings to light one of the sources of the difficulty of defining theological infinity as the lack of limits. In this connexion, it seems that there are always two trends in theology: one that wants to make sense out of God's infinity and one that wants to preserve God's infinity as an unexplored &mdash and unexplorable &mdash mystery. The definition of `unlimited' and the attitude towards the mystery of the infinite are interwoven. Essentially, the point is that there is no agreement about the extent to which God is unlimited, and hence no agreement on the content of the concept of God's infinite goodness, infinite power and so on.

Can God contravene the laws of logic? Can God do evil so that a greater good will come? We can answer these questions either way, and the meaning of `unlimited' will vary as the answers vary.

Ivan Karamazov wants an explanation of evil that will be understandable for his `finite Euclidean mind': the only answer he gets says that understanding is impossible and the most he could attain would be an acceptance stemming from mystical love that somehow transcends his own limits. Here is one of the points where the mathematical concept of infinity cannot help us clarify the concept of theological infinity. There are these two trends in theology: one that wants an answer to the problems that seem to come from God's infinity and one that insists upon the mystery of the infinite. These are not trends that you find in mathematics. And in mathematics a contradiction indicates error: in theology, however, a contradiction can be taken also as indicating that the subject matter lies beyond our grasp.

It seems that the development of the mathematical concept of the infinite has gone on increasingly independently of the theological concept of the infinite. The mathematical infinite has become, particularly during this century, an ordinary, unmysterious operational concept. The theological infinite, however, retains its connexion with mystery. What Bochner calls the "secularization" of infinity has taken place within the realm of the mathematical. In mathematics, there may not be universal agreement about the philosophical meaning of infinity, but there is at least agreement about methods and goals and there are means of determining, to some extent, the suitability of conceptions of the infinite. In theology, there is no such agreement.

A not uncommon answer to theological difficulties is to say that the difficulties stem from God's infinity: we cannot grasp the infinite, so we cannot answer the questions. But mathematics shows that, while we may not be able to encompass the infinite &mdash we cannot actually pass through all the numbers in an infinite series &mdash we can still grasp the infinite, and by finite means. Mathematical concepts like `addition' can be applied to both finite and infinite collections. Certain concepts have been reworked so as to encompass both finite and transfinite operations the concepts of finite arithmetic do not automatically carry over into transfinite arithmetic. For instance `addition', although it is a concept that is applicable to both the finite and the transfinite, does not behave in the same way in both cases: the property of commutativity, for instance, is not assured for the addition of transfinite ordinals. A similar sort of restructuring of the concepts seems not to be possible for theology. As long as there is some insistence on preserving the mystery of the theological infinite, any agreement about restructuring concepts to encompass the infinite is prevented. Moreover, there are standards in mathematics for this restructuring of concepts &mdash whether the extended concepts work. In theology, though, there is not even any agreement about whether concepts applied to God ought to `work'. If God essentially transcends our knowledge, then there is no reason to believe that the concepts we apply to God should `make sense'.

At this point we are faced with grave difficulties in defining the theological conception of the infinite. The theological conception of the infinite is defined by Descartes and Leibniz as a lack of limits. But there is no agreement about nor, apparently, any means of determining, how far this lack of limits reaches and what effect it has on predication. Furthermore, the conception of the unlimited is, according to mathematics, strictly independent from the infinite. And we have to ask how infinity could be an explanation for the difficulty in extending predicates from the mundane to the absolute, when there is no comparable degree of difficulty encountered in extending concepts from the finite to the infinite in mathematics. The link between the mathematical and the theological concepts of infinity seems at this point so tenuous that we might wonder whether `infinity', used of God, is not simply equivocal.

2.5 Infinity as metaphor

We might explore the possibility of a figurative attribution of infinity to God. Saying that God is infinite is meant to suggest that God shares the unending and mysterious quality of the infinite. Even the mathematical infinite remains a fascinating and compelling concept. Escher's pictures and some of Borges's stories, though based on aspects of mathematical infinity, have a power to cause delight or anxiety. Maybe it is this power, present even in the rationally defined concept, that is most relevant. Clearly, there is such a figurative use of infinity. I might say that something is infinitely beautiful to me: by this I should mean that I could never get tired of looking at the object and that its beauty comes from some as it were inexhaustible and mysterious source &mdash mysterious in virtue of the fact that I cannot fully understand or articulate the reasons for the object's power over me. `Infinitely' here incorporates the mystery and the inexhaustibility, as well as suggesting a scope or a depth beyond the ordinary.

Perhaps there is a development of `infinite' as an attribute of God from the literal to the figurative. Before all the developments in mathematics &mdash before our own century particularly, and before the seventeenth century altogether &mdash the mathematical concept of the infinite was as tied to mystery as is the theological concept. In that context it was not senseless to say that God is literally infinite. But now, the mathematical concept of the infinite, and mathematical methods for dealing with it, make it impossible to suggest that God's infinity is the reason for any difficulties in thinking or talking about God. We cannot, now, simply stand in awe of the infinite and claim that it is not like anything else &mdash mathematics shows that the infinite is in many respects not all that unlike the finite.

What we may be seeing here is a case where the term was originally used literally, and then retreated to a metaphorical status, because of the development and `secularization' of the infinite. Another possibility is that theological infinity has always been a mathematical metaphor conveying mystery and immensity, but that the metaphor has retained a meaning that has gradually been lost on its home ground. (Similar, perhaps, to the exemplar `Man is a wolf', which can be used to convey characteristics no longer believed to be possessed by wolves.)

The mathematical infinite cannot provide a direct, positive clarification of the meaning of the theological infinite. What the concept of the infinite can express about God does not lie in the gradual mathematical demystifying of the concept, nor even in the residual strange bits of the mathematical infinite, but rather in the wonder (or dread) we have felt in being introduced to infinite series or set theory, or in looking at Escher's pictures, or at the sky at night. Infinity fascinates us. The meaning of the metaphor of the infinite, as opposed to the useful operational infinite, is no more (or no less) than the mystery and wonder attached to the incomprehensibly immense. It may be that there is, historically, a literal source common to the mathematical and the theological infinite, but these concepts have diverged as the mathematical meaning has become more clear and less mysterious.

2.6 Incommensurability

We might still claim to find in infinity a reason for difficulty in attributing concepts to God, in virtue of the fact that the theological infinite is incommensurable with the finite as well as with the potential and actual infinities. As I am using `incommensurability' here, incommensurability can be said to involve an insurpassable gap, a discontinuity, the suspension of addition . . . we have a positive value that, no matter how often a certain amount is added to itself, cannot become greater than another positive value, and cannot . . . because they are the sort of value that, even remaining constant, cannot add up to some other value. 12 [James Griffin, Well-Being (Oxford: Clarendon Press, 1986), pp. 85ff.]

Whatever the theological infinite is, it is not constructible from the finite &mdash not in fact and not in principle. We cannot get there from here there is no process by which we would get to the theological infinite. This has been emphasized in the discussion of Descartes and Leibniz above. And we may wonder whether this insurpassable gap means that whatever measure we use for the finite and the `mundane' infinite, is not a measure for the theological infinite.

We could try saying that the same concepts cannot apply both to the finite and to the theological infinite, because of this incommensurability. The same concepts can, however, apply both to the finite and to the actual or potential infinite, because these types of infinite are constructible, at least in principle, from the finite. And then the fact that concepts stretch from the finite to the infinite in mathematics would not rule out the possibility of infinity's being an explanation for expressibility problems in theology.

A relevant point here is the idea that, while we know that God is good, we do not know what it is for God to be good: we do not know the truth-conditions or criteria for God's goodness. According to this idea, just adding on to the mundane concept of goodness will not get us to a real understanding of God's goodness. God's goodness is not just bigger, so to speak, than human goodness it is different, on a different scale altogether. If I, for instance, were to have power added to me, even to infinity, I can imagine I should be able to move large pieces of furniture without help, work for hours and hours, and so forth. But this is not what `infinite power' means for God. If I had infinite power, I should have infinite human power I should still not be able to create something in its being ex nihilo. Again, the point is that God's power is not limited by the nature of God, as my power is limited by my (human) nature. We might wonder if there is a sort of conceptual gap here: while we can imagine what it would be like (give the truth-conditions) for a mundane quality to be increased to infinity, we cannot grasp what is infinite on another scale altogether.

This line of thought, though, is also questionable. It need not be impossible to grasp what is infinite on another scale altogether it may simply be that we have not yet found the scale, not found a useful model. Even if two things are incommensurable in the sense defined, it need not follow that the same concepts do not apply to both. Certainly my intellect is incommensurable with Leibniz's in this sense: no matter how many of me were working together, we should in all likelihood not have invented calculus &mdash but the same concepts of intellect apply to both me and Leibniz.

It seems that incommensurability, too, can be seen as a figurative expression of God's transcendence. God's infinity is said to transcend mundane infinity, the sort of infinity that we can construct and grasp, not because we are barred from reaching the theological infinite by some practical considerations, as someone might say that we are barred from reaching the actual infinite. Rather, we cannot reach the theological infinite because we have no process for constructing it, no means of reaching it.

III. CONCLUSION

If `infinite', used of God, is a figurative expression of mystery, immensity, and transcendence, then it is not a cause or reason for problems with expressibility. `Absolutely infinite' does not give us a new and useful piece of knowledge about God, the way that `denumerably infinite' does give us new and useful knowledge about the rational numbers. It is not so much that we `discover' that God is infinite, and thereby discover a good explanation for the difficulties of predication that we have experienced. We might just as well say that, experiencing God as mysteriously unlimited and, so to speak, unimaginably large, we attribute to God a term that expresses &mdash among other things &mdash wonderful and fascinating mystery and inexhaustibility in other matters.

It may be that the proposition `God is infinite' can be taken literally only when infinity in general is shrouded in mystery. When infinity is `secularized' and mathematical infinity is given an increasingly clear and unmysterious meaning, `infinite' can be understood of God only figuratively. In this figurative meaning, `infinite' is meant to suggest the ever-present fascination and, as it were, unwieldy vastness of the infinite. We cannot somehow extend the mathematical concept of the infinite in order to gain a deeper understanding of the nature of God. Instead, comparison of the mathematical with the theological concept brings into light the experience of God as the mysterious unspeakable. 13 [I am grateful to Hans Herzberger for his comments on earlier drafts of this material. I am also grateful to Edwin Mares for his suggestions.]

Department of Philosophy,
McMaster University,
Hamilton, Ontario,
Canada, L8S 4K1


Contents

The first four hyperoperations are shown here, with tetration being considered the fourth in the series. The unary operation succession, defined as a ′ = a + 1 , is considered to be the zeroth operation.

Succession, (a′ = a + 1) , is the most basic operation while addition ( a + n ) is a primary operation, for addition of natural numbers it can be thought of as a chained succession of n successors of a multiplication (a × n ) is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving n numbers of a . Exponentiation can be thought of as a chained multiplication involving n numbers of a and tetration ( n a a!> ) as a chained power involving n numbers a . Each of the operations above are defined by iterating the previous one [2] however, unlike the operations before it, tetration is not an elementary function.

The parameter a is referred to as the base, while the parameter n may be referred to as the height. In the original definition of tetration, the height parameter must be a natural number for instance, it would be illogical to say "three raised to itself negative five times" or "four raised to itself one half of a time." However, just as addition, multiplication, and exponentiation can be defined in ways that allow for extensions to real and complex numbers, several attempts have been made to generalize tetration to negative numbers, real numbers, and complex numbers. One such way for doing so is using a recursive definition for tetration for any positive real a > 0 and non-negative integer n ≥ 0 , we can define n a a>> recursively as: [2]

The recursive definition is equivalent to repeated exponentiation for natural heights however, this definition allows for extensions to the other heights such as 0 a a> , − 1 a a> , and i a a> as well – many of these extensions are areas of active research.

There are many terms for tetration, each of which has some logic behind it, but some have not become commonly used for one reason or another. Here is a comparison of each term with its rationale and counter-rationale.

  • The term tetration, introduced by Goodstein in his 1947 paper Transfinite Ordinals in Recursive Number Theory[3] (generalizing the recursive base-representation used in Goodstein's theorem to use higher operations), has gained dominance. It was also popularized in Rudy Rucker's Infinity and the Mind.
  • The term superexponentiation was published by Bromer in his paper Superexponentiation in 1987. [4] It was used earlier by Ed Nelson in his book Predicative Arithmetic, Princeton University Press, 1986.
  • The term hyperpower[5] is a natural combination of hyper and power, which aptly describes tetration. The problem lies in the meaning of hyper with respect to the hyperoperation sequence. When considering hyperoperations, the term hyper refers to all ranks, and the term super refers to rank 4, or tetration. So under these considerations hyperpower is misleading, since it is only referring to tetration.
  • The term power tower[6] is occasionally used, in the form "the power tower of order n " for a a ⋅ ⋅ a ⏟ n ><>>> > atop n>> . This is a misnomer, however, because tetration cannot be expressed with iterated power functions (see above), since it is an iterated exponential function.

Owing in part to some shared terminology and similar notational symbolism, tetration is often confused with closely related functions and expressions. Here are a few related terms:

Terms related to tetration
Terminology Form
Tetration a a ⋅ ⋅ a a >>>>
Iterated exponentials a a ⋅ ⋅ a x <>>>>>>
Nested exponentials (also towers) a 1 a 2 ⋅ ⋅ a n ^^>>>>>
Infinite exponentials (also towers) a 1 a 2 a 3 ⋅ ⋅ ⋅ ^^^>>>>>

In the first two expressions a is the base, and the number of times a appears is the height (add one for x ). In the third expression, n is the height, but each of the bases is different.

Care must be taken when referring to iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as this can either mean iterated powers or iterated exponentials.

There are many different notation styles that can be used to express tetration. Some notations can also be used to describe other hyperoperations, while some are limited to tetration and have no immediate extension.

Notation styles for tetration
Name Form Description
Rudy Rucker notation n a ^a> Used by Maurer [1901] and Goodstein [1947] Rudy Rucker's book Infinity and the Mind popularized the notation. [nb 1]
Knuth's up-arrow notation a ↑ ↑ n n> Allows extension by putting more arrows, or, even more powerfully, an indexed arrow.
Conway chained arrow notation a → n → 2 Allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain
Ackermann function n 2 = A ⁡ ( 4 , n − 3 ) + 3 ^2=operatorname (4,n-3)+3> Allows the special case a = 2 to be written in terms of the Ackermann function.
Iterated exponential notation exp a n ⁡ ( 1 ) (1)> Allows simple extension to iterated exponentials from initial values other than 1.
Hooshmand notations [7] uxp a ⁡ n a n &operatorname _n[2pt]&a^<>>end>> Used by M. H. Hooshmand [2006].
Hyperoperation notations a [ 4 ] n H 4 ( a , n ) &a[4]n[2pt]&H_<4>(a,n)end>> Allows extension by increasing the number 4 this gives the family of hyperoperations.
Double caret notation a^^n Since the up-arrow is used identically to the caret ( ^ ), tetration may be written as ( ^^ ) convenient for ASCII.

One notation above uses iterated exponential notation this is defined in general as follows:

There are not as many notations for iterated exponentials, but here are a few:

Notation styles for iterated exponentials
Name Form Description
Standard notation exp a n ⁡ ( x ) (x)> Euler coined the notation exp a ⁡ ( x ) = a x > , and iteration notation f n ( x ) (x)> has been around about as long.
Knuth's up-arrow notation ( a ↑ ) n ( x ) )^(x)> Allows for super-powers and super-exponential function by increasing the number of arrows used in the article on large numbers.
Text notation exp _ a ^ n(x) Based on standard notation convenient for ASCII.
J Notation x ^^: ( n - 1 ) x Repeats the exponentiation. See J (programming language) [8]

Because of the extremely fast growth of tetration, most values in the following table are too large to write in scientific notation. In these cases, iterated exponential notation is used to express them in base 10. The values containing a decimal point are approximate.

Tetration has several properties that are similar to exponentiation, as well as properties that are specific to the operation and are lost or gained from exponentiation. Because exponentiation does not commute, the product and power rules do not have an analogue with tetration the statements a ( b x ) = ( a b x ) < extstyle <>^left(<>^x ight)=left(<>^x ight)> and a ( x y ) = a x a y < extstyle <>^left(xy ight)=<>^x<>^y> are not necessarily true for all cases. [9]

When a number x and 10 are coprime, it is possible to compute the last m decimal digits of a x using Euler's theorem, for any integer m .

Direction of evaluation Edit

When evaluating tetration expressed as an "exponentiation tower", the serial exponentiation is done at the deepest level first (in the notation, at the apex). [1] For example:

This order is important because exponentiation is not associative, and evaluating the expression in the opposite order will lead to a different answer:

Evaluating the expression the left to right is considered less interesting evaluating left to right, any expression n a a!> can be simplified to be a ( a n − 1 ) ight)>!!> . [10] Because of this, the towers must be evaluated from right to left (or top to bottom). Computer programmers refer to this choice as right-associative.


Math Insight

In simple cases, one can calculate a limit by just pluggin in the limit value. But sometimes things &lsquoblow up&rsquo when the limit number is substituted: $lim_ =<0over 0>. $ Ick. This is not good. However, in this example, as in many examples, doing a bit of simplifying algebra first gets rid of the factors in the numerator and denominator which cause them to vanish: $lim_ = lim_ <(x-3)(x+3)over x-3>= lim_ <(x+3)over 1>=<(3+3)over 1>=6$ Here at the very end we did just plug in, after all.

The lesson here is that some of those darn algebra tricks (&lsquoidentities&rsquo) are helpful, after all. If you have a &lsquobad&rsquo limit, always look for some cancellation of factors in the numerator and denominator.

In fact, for hundreds of years people only evaluated limits in this style! After all, human beings can't really execute infinite limiting processes, and so on.


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