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1.4: One Sided Limits - Mathematics


We introduced the concept of a limit gently, approximating their values graphically and numerically. Next came the rigorous definition of the limit, along with an admittedly tedious method for evaluating them. The previous section gave us tools (which we call theorems) that allow us to compute limits with greater ease. Chief among the results were the facts that polynomials and rational, trigonometric, exponential and logarithmic functions (and their sums, products, etc.) all behave "nicely.'' In this section we rigorously define what we mean by "nicely.''

In Section 1.1 we explored the three ways in which limits of functions failed to exist:

  1. The function approached different values from the left and right,
  2. The function grows without bound, and
  3. The function oscillates.

In this section we explore in depth the concepts behind #1 by introducing the one-sided limit. We begin with formal definitions that are very similar to the definition of the limit given in Section 1.2, but the notation is slightly different and "(x eq c)'' is replaced with either "(xc).''

Definition 2: One Sided Limits

Left-Hand Limit

Let (I) be an open interval containing (c), and let (f) be a function defined on (I), except possibly at (c). The limit of (f(x)), as (x) approaches (c) from the left, is (L), or, the left--hand limit of (f) at (c) is (L), denoted by

[ limlimits_{x ightarrow c^-} f(x) = L,]

means that given any (epsilon > 0), there exists (delta > 0) such that for all (x< c), if (|x - c| < delta), then (|f(x) - L| < epsilon).

Right-Hand Limit

Let (I) be an open interval containing (c), and let (f) be a function defined on (I), except possibly at (c). The limit of (f(x)), as (x) approaches (c) from the right, is (L), or, the right--hand limit of (f) at (c) is (L), denoted by

[ limlimits_{x ightarrow c^+} f(x) = L,]

means that given any (epsilon > 0), there exists (delta > 0) such that for all (x> c), if (|x - c| < delta), then (|f(x) - L| < epsilon).

Practically speaking, when evaluating a left-hand limit, we consider only values of (x) "to the left of (c),'' i.e., where (xc). We can use the theorems from previous sections to help us evaluate these limits; we just restrict our view to one side of (c).

We practice evaluating left and right-hand limits through a series of examples.

Example 17: Evaluating one sided limits

Let ( f(x) = left{egin{array}{cc} x & 0leq xleq 1 3-x & 1

  1. (limlimits_{x o 1^-} f(x))
  2. (limlimits_{x o 1^+} f(x))
  3. (limlimits_{x o 1} f(x))
  4. (f(1))
  5. (limlimits_{x o 0^+} f(x) )
  6. (f(0))
  7. (limlimits_{x o 2^-} f(x))
  8. (f(2))


( ext{FIGURE 1.21}): A graph of (f) in Example 17.

Solution

For these problems, the visual aid of the graph is likely more effective in evaluating the limits than using (f) itself. Therefore we will refer often to the graph.

  1. As (x) goes to 1 from the left, we see that (f(x)) is approaching the value of 1. Therefore ( limlimits_{x o 1^-} f(x) =1.)
  2. As (x) goes to 1 from the right, we see that (f(x)) is approaching the value of 2. Recall that it does not matter that there is an "open circle'' there; we are evaluating a limit, not the value of the function. Therefore ( limlimits_{x o 1^+} f(x)=2).
  3. The limit of (f) as (x) approaches 1 does not exist, as discussed in the first section. The function does not approach one particular value, but two different values from the left and the right.
  4. Using the definition and by looking at the graph we see that (f(1) = 1).
  5. As (x) goes to 0 from the right, we see that (f(x)) is also approaching 0. Therefore ( limlimits_{x o 0^+} f(x)=0). Note we cannot consider a left-hand limit at 0 as (f) is not defined for values of (x<0).
  6. Using the definition and the graph, (f(0) = 0).
  7. As (x) goes to 2 from the left, we see that (f(x)) is approaching the value of 1. Therefore ( limlimits_{x o 2^-} f(x)=1.)
  8. The graph and the definition of the function show that (f(2)) is not defined.

Note how the left and right-hand limits were different at (x=1). This, of course, causes the limit to not exist. The following theorem states what is fairly intuitive: the limit exists precisely when the left and right-hand limits are equal.

Theorem 7: Limits and One Sided Limits

Let (f) be a function defined on an open interval (I) containing (c). Then [limlimits_{x o c}f(x) = L]if, and only if, [limlimits_{x o c^-}f(x) = L quad ext{and} quad limlimits_{x o c^+}f(x) = L.]

The phrase "if, and only if'' means the two statements are equivalent: they are either both true or both false. If the limit equals (L), then the left and right hand limits both equal (L). If the limit is not equal to (L), then at least one of the left and right-hand limits is not equal to (L) (it may not even exist).

One thing to consider in Examples 17 - 20 is that the value of the function may/may not be equal to the value(s) of its left/right-hand limits, even when these limits agree.

Example 18: Evaluating limits of a piecewise-defined function

Let (f(x) = left{egin{array}{cc} 2-x & 0

  1. ( limlimits_{x o 1^-} f(x))
  2. ( limlimits_{x o 1^+} f(x))
  3. ( limlimits_{x o 1} f(x))
  4. ( f(1))
  5. ( limlimits_{x o 0^+} f(x))
  6. (f(0))
  7. ( limlimits_{x o 2^-} f(x))
  8. (f(2))


( ext{FIGURE 1.22}): A graph of (f) from Example 18.

Solution

Again we will evaluate each using both the definition of (f) and its graph.

  1. As (x) approaches 1 from the left, we see that (f(x)) approaches 1. Therefore ( limlimits_{x o 1^-} f(x)=1.)
  2. As (x) approaches 1 from the right, we see that again (f(x)) approaches 1. Therefore ( limlimits_{x o 1+} f(x)=1).
  3. The limit of (f) as (x) approaches 1 exists and is 1, as (f) approaches 1 from both the right and left. Therefore ( limlimits_{x o 1} f(x)=1).
  4. (f(1)) is not defined. Note that 1 is not in the domain of (f) as defined by the problem, which is indicated on the graph by an open circle when (x=1).
  5. As (x) goes to 0 from the right, (f(x)) approaches 2. So ( limlimits_{x o 0^+} f(x)=2).
  6. (f(0)) is not defined as (0) is not in the domain of (f).
  7. As (x) goes to 2 from the left, (f(x)) approaches 0. So ( limlimits_{x o 2^-} f(x)=0).
  8. (f(2)) is not defined as 2 is not in the domain of (f).

Example 19: Evaluating limits of a piecewise-defined function

Let (f(x) = left{egin{array}{cc} (x-1)^2 & 0leq xleq 2, x eq 1 1 & x=1end{array}, ight.) as shown in Figure 1.23. Evaluate the following.

  1. ( limlimits_{x o 1^-} f(x))
  2. ( limlimits_{x o 1^+} f(x))
  3. ( limlimits_{x o 1} f(x))
  4. (f(1))


( ext{FIGURE 1.23}): Graphing (f) in Example 19.

It is clear by looking at the graph that both the left and right-hand limits of (f), as (x) approaches 1, is 0. Thus it is also clear that the limit is 0; i.e., ( limlimits_{x o 1} f(x) = 0). It is also clearly stated that (f(1) = 1).

Example 20: Evaluating limits of a piecewise-defined function

Let (f(x) = left{egin{array}{cc} x^2 & 0leq xleq 1 2-x & 1

  1. ( limlimits_{x o 1^-} f(x))
  2. ( limlimits_{x o 1^+} f(x))
  3. ( limlimits_{x o 1} f(x))
  4. (f(1))


( ext{FIGURE 1.24}): Graphing (f) in Example 20.

Solution

It is clear from the definition of the function and its graph that all of the following are equal:

[limlimits_{x o 1^-} f(x) = limlimits_{x o 1^+} f(x) =limlimits_{x o 1} f(x) =f(1) = 1.]

In Examples 17 - 20 we were asked to find both ( limlimits_{x o 1}f(x)) and (f(1)). Consider the following table:

[egin{array}{ccc} & limlimits_{x o 1}f(x) & f(1) hline ext{Example 17} & ext{does not exist} & 1 ext{Example 18} & 1 & ext{not defined} ext{Example 19} & 0 & 1 ext{Example 20} & 1 & 1 end{array}]

Only in Example 20 do both the function and the limit exist and agree. This seems "nice;'' in fact, it seems "normal.'' This is in fact an important situation which we explore in the next section, entitled "Continuity.'' In short, a continuous function is one in which when a function approaches a value as (x ightarrow c) (i.e., when ( limlimits_{x o c} f(x) = L)), it actually attains that value at (c). Such functions behave nicely as they are very predictable.


How to Solve One-Sided Limits

Use the graph to approximate the value of both one-sided limits as $x$ approaches 3.

Examine what happens as $x$ approaches from the left.

As $x$ approaches 3 from the left, the function seems to be approaching 2.

Examine what happens as x approaches from the right

As $x$ approaches 3 from the right, the function seems to be approaching 3.

Example 2: Using Tables

Using the tables below, what can be said about the one-sided limits as $x$ approaches 6?

$ egin x & f(x)hline hline 5 & 8.32571hline 5.5 & 8.95692hline 5.9 & 8.99084hline 5.99 & 8.99987hline 5.999 & 8.99992hline 5.9999 & 8.99999hline end $

$ egin x & f(x)hline hline 7 & 1hline 6.5 & 95hline 6.1 & 1230hline 6.01 & 9658hline 6.001 & 54231hline 6.0001 & 834366 end $

Examine what happens as $x$ approaches 6 from the left.

As $x$ approaches 6 from the left.

$ egin x & f(x)hline hline 5 & 8.32571hline 5.5 & 8.95692hline 5.9 & 8.99084hline 5.99 & 8.99987hline 5.999 & 8.99992hline 5.9999 & 8.99999 end $

. the function seems to be getting closer to 9.

Examine what happens as $x$ approaches 6 from the right.

As $x$ approaches 6 from the left.

$ egin x & f(x)hline hline 7 & 1hline 6.5 & 95hline 6.1 & 1230hline 6.01 & 9658hline 6.001 & 54231hline 6.0001 & 834366 end $

. the function seems to just keep getting bigger.

Practice Problems

Use the graph below to find the limits in questions 1--4.

Problem 1

$displaystyle lim_ f(x) approx$


One-Sided Continuity. Classification of Discontinuities

Similarly to the one-sided limits, we can define one-sided continuity.

Clearly, if function is continuous from the left and from the right at point $$ $$ , then it is continuous at point $$ $$ .

Definition. Function $$ > $$ is discontinuous at $$ $$ if it is not continuous.

There are three kinds of discontinuity at $$ $$ :

Let's go through a couple of examples of these discontinuities.

Example 1 . Find where function $$ <> ight)>>>=frac<<<>^<<2>>+<2>-<3>>><<-<1>>> $$ is discontinuous and classify this discontinuity.

This function is rational, so it is continuous everywhere, except where denominator equals 0, i.e. where $$ -<1>= <0>$$ . So, functions is not defined (and not continuous) when $$ = <1>$$ .

Thus, limit exists and finite, but $$ ight)>>> $$ is not defined, so $$ = <1>$$ is removable discontinuity.

Example 2 . Find where function $$ <> ight)>>>=frac<<<>^<<2>>+<2>-<3>>><<-<1>>>quad> e<1><2>quad>=<1> end ight.> $$ is discontinuous and classify this discontinuity.

This is actually same example as example 1, except that function is defined at $$ = <1>$$ .

Thus, $$ = <1>$$ is point of removable discontinuity.

Example 3 . Find where function $$ <> ight)>>>=+<1>quad>le<2><4>-quad>><2> end ight.> $$ is discontinuous and classify this discontinuity.

This means that at $$ = <2>$$ there is jump discontinuity.

Also note, that $$ lim_<< o<<2>>^<<->>>><> ight)>>>=<3>= ight)>>> $$ , this means that function is continuous from the left at $$ = <2>$$ .

Example 4 . Find where function $$ <> ight)>>>=frac<<1>><<-<1>>> $$ is discontinuous and classify this discontinuity.

This function is rational, so it is continuous everywhere, except where denominator equals 0, i.e. where $$ -<1>=<0>. $$ So, functions is not continuous when $$ = <1>$$ .

This means that at $$ = <1>$$ there is infinite discontinuity.

Actually infinite discontinuity occurs when we have vertical asymptote.

Example 5 . Find points where function $$ > $$ is discontinuous and classify these points: $$ <> ight)>>>=frac<<1>><>quad><<2>frac<<1>><>quad><2><le<4>quad><4><<<6><4>quad>=<6>\quad>><6> end ight.> $$

Function is discontinuous at 0 because $$ lim_<< o<<0>>^<+>>><> ight)>>>=lim_<< o<<0>>^<+>>>frac<<1>><>=infty $$ . So, at function $$ = <0>$$ we have infinite discontinuity.

Function is dicontinuous at 2 because $$ ight)>>> $$ is simply not defined. Since $$ lim_<< o<<2>>^<<->>>>frac<<1>><>=lim_<< o<<2>>^<+>>>frac<<1>><>=frac<<1>><<2>> $$ then $$ lim_<< o<2>>>frac<<1>><>=frac<<1>><<2>> $$ and so $$ = <2>$$ is removable discontinuity.

This means that $$ = <4>$$ is jump discontinuity.

This means that $$ = <6>$$ is removable discontinuity.

Example 6 . Conside function $$ <> ight)>>>= ight]> $$ where $$ ight]> $$ is a floor function.

This function is discontinuous at every integer point $$ = $$ , because $$ lim_<< o<>^<<->>>> ight]>=- <1>$$ and $$ lim_<< o<>^<+>>> ight]>= $$ .

So one-sided limits are finite but not equal. This means that at every integer point $$ = $$ there is jump discontinuity.

Also, note that $$ <> ight)>>>==lim_<< o<>^<+>>><> ight)>>> $$ that's why at every integer point $$ = $$ function is continuous from the right.


Contents

The limit inferior of a sequence (xn) is defined by

Similarly, the limit superior of (xn) is defined by

If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as the real numbers together with ±∞ (i. e. the extended real number line) are complete. More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice.

Whenever the ordinary limit exists, the limit inferior and limit superior are both equal to it therefore, each can be considered a generalization of the ordinary limit which is primarily interesting in cases where the limit does not exist. Whenever lim inf xn and lim sup xn both exist, we have

Limits inferior/superior are related to big-O notation in that they bound a sequence only "in the limit" the sequence may exceed the bound. However, with big-O notation the sequence can only exceed the bound in a finite prefix of the sequence, whereas the limit superior of a sequence like e −n may actually be less than all elements of the sequence. The only promise made is that some tail of the sequence can be bounded above by the limit superior plus an arbitrarily small positive constant, and bounded below by the limit inferior minus an arbitrarily small positive constant.

The limit superior and limit inferior of a sequence are a special case of those of a function (see below).

In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers. Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the complete totally ordered set [−∞,∞], which is a complete lattice.

Interpretation Edit

Properties Edit

The relationship of limit inferior and limit superior for sequences of real numbers is as follows:

If I = lim inf n → ∞ x n x_> and S = lim sup n → ∞ x n x_> , then the interval [I, S] need not contain any of the numbers xn, but every slight enlargement [I − ε, S + ε] (for arbitrarily small ε > 0) will contain xn for all but finitely many indices n. In fact, the interval [I, S] is the smallest closed interval with this property. We can formalize this property like this: there exist subsequences x k n >> and x h n >> of x n > (where k n > and h n > are monotonous) for which we have

The liminf and limsup of a sequence are respectively the smallest and greatest cluster points.

  • For any two sequences of real numbers < a n >, < b n >>,<>>> , the limit superior satisfies subadditivity whenever the right side of the inequality is defined (i.e., not ∞ − ∞ or − ∞ + ∞ ):

Analogously, the limit inferior satisfies superadditivity:

hold whenever the right-hand side is not of the form 0 ⋅ ∞ .

Examples Edit

  • As an example, consider the sequence given by xn = sin(n). Using the fact that pi is irrational, one can show that

(This is because the sequence <1,2,3. >is equidistributed mod 2π, a consequence of the Equidistribution theorem.)

where pn is the n-th prime number. The value of this limit inferior is conjectured to be 2 – this is the twin prime conjecture – but as of April 2014 [update] has only been proven to be less than or equal to 246. [2] The corresponding limit superior is + ∞ , because there are arbitrary gaps between consecutive primes.

Assume that a function is defined from a subset of the real numbers to the real numbers. As in the case for sequences, the limit inferior and limit superior are always well-defined if we allow the values +∞ and -∞ in fact, if both agree then the limit exists and is equal to their common value (again possibly including the infinities). For example, given f(x) = sin(1/x), we have lim supx0 f(x) = 1 and lim infx0 f(x) = -1. The difference between the two is a rough measure of how "wildly" the function oscillates, and in observation of this fact, it is called the oscillation of f at 0. This idea of oscillation is sufficient to, for example, characterize Riemann-integrable functions as continuous except on a set of measure zero. [3] Note that points of nonzero oscillation (i.e., points at which f is "badly behaved") are discontinuities which, unless they make up a set of zero, are confined to a negligible set.

There is a notion of lim sup and lim inf for functions defined on a metric space whose relationship to limits of real-valued functions mirrors that of the relation between the lim sup, lim inf, and the limit of a real sequence. Take a metric space X, a subspace E contained in X, and a function f : ER. Define, for any limit point a of E,

where B(aε) denotes the metric ball of radius ε about a.

Note that as ε shrinks, the supremum of the function over the ball is monotone decreasing, so we have

This finally motivates the definitions for general topological spaces. Take X, E and a as before, but now let X be a topological space. In this case, we replace metric balls with neighborhoods:

(there is a way to write the formula using "lim" using nets and the neighborhood filter). This version is often useful in discussions of semi-continuity which crop up in analysis quite often. An interesting note is that this version subsumes the sequential version by considering sequences as functions from the natural numbers as a topological subspace of the extended real line, into the space (the closure of N in [−∞,∞], the extended real number line, is N ∪ <∞>.)

The power set ℘(X) of a set X is a complete lattice that is ordered by set inclusion, and so the supremum and infimum of any set of subsets (in terms of set inclusion) always exist. In particular, every subset Y of X is bounded above by X and below by the empty set ∅ because ∅ ⊆ YX. Hence, it is possible (and sometimes useful) to consider superior and inferior limits of sequences in ℘(X) (i.e., sequences of subsets of X).

There are two common ways to define the limit of sequences of sets. In both cases:

  • The sequence accumulates around sets of points rather than single points themselves. That is, because each element of the sequence is itself a set, there exist accumulation sets that are somehow nearby to infinitely many elements of the sequence.
  • The supremum/superior/outer limit is a set that joins these accumulation sets together. That is, it is the union of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it contains each of them. Hence, it is the supremum of the limit points.
  • The infimum/inferior/inner limit is a set where all of these accumulation sets meet. That is, it is the intersection of all of the accumulation sets. When ordering by set inclusion, the infimum limit is the greatest lower bound on the set of accumulation points because it is contained in each of them. Hence, it is the infimum of the limit points.
  • Because ordering is by set inclusion, then the outer limit will always contain the inner limit (i.e., lim inf Xn ⊆ lim sup Xn). Hence, when considering the convergence of a sequence of sets, it generally suffices to consider the convergence of the outer limit of that sequence.

The difference between the two definitions involves how the topology (i.e., how to quantify separation) is defined. In fact, the second definition is identical to the first when the discrete metric is used to induce the topology on X.

General set convergence Edit

In this case, a sequence of sets approaches a limiting set when the elements of each member of the sequence approach the elements of the limiting set. In particular, if <Xn> is a sequence of subsets of X, then:

  • lim sup Xn, which is also called the outer limit, consists of those elements which are limits of points in Xn taken from (countably) infinitely many n. That is, x ∈ lim sup Xn if and only if there exists a sequence of points xk and a subsequence<Xnk> of <Xn> such that xkXnk and xkx as k → ∞.
  • lim inf Xn, which is also called the inner limit, consists of those elements which are limits of points in Xn for all but finitely many n (i.e., cofinitely many n). That is, x ∈ lim inf Xn if and only if there exists a sequence of points <xk> such that xkXk and xkx as k → ∞.

The limit lim Xn exists if and only if lim inf Xn and lim sup Xn agree, in which case lim Xn = lim sup Xn = lim inf Xn. [4]

Special case: discrete metric Edit

This is the definition used in measure theory and probability. Further discussion and examples from the set-theoretic point of view, as opposed to the topological point of view discussed below, are at set-theoretic limit.

By this definition, a sequence of sets approaches a limiting set when the limiting set includes elements which are in all except finitely many sets of the sequence and does not include elements which are in all except finitely many complements of sets of the sequence. That is, this case specializes the general definition when the topology on set X is induced from the discrete metric.

Specifically, for points xX and yX, the discrete metric is defined by

under which a sequence of points <xk> converges to point xX if and only if xk = x for all except finitely many k. Therefore, if the limit set exists it contains the points and only the points which are in all except finitely many of the sets of the sequence. Since convergence in the discrete metric is the strictest form of convergence (i.e., requires the most), this definition of a limit set is the strictest possible.

If <Xn> is a sequence of subsets of X, then the following always exist:

  • lim sup Xn consists of elements of X which belong to Xn for infinitely manyn (see countably infinite). That is, x ∈ lim sup Xn if and only if there exists a subsequence <Xnk> of <Xn> such that xXnk for all k.
  • lim inf Xn consists of elements of X which belong to Xn for all except finitely manyn (i.e., for cofinitely many n). That is, x ∈ lim inf Xn if and only if there exists some m>0 such that xXn for all n>m.

Observe that x ∈ lim sup Xn if and only if x ∉ lim inf Xn c .

  • The lim Xn exists if and only if lim inf Xn and lim sup Xn agree, in which case lim Xn = lim sup Xn = lim inf Xn.

In this sense, the sequence has a limit so long as every point in X either appears in all except finitely many Xn or appears in all except finitely many Xn c . [5]

Using the standard parlance of set theory, set inclusion provides a partial ordering on the collection of all subsets of X that allows set intersection to generate a greatest lower bound and set union to generate a least upper bound. Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf Xn, is the largest meeting of tails of the sequence, and the outer limit, lim sup Xn, is the smallest joining of tails of the sequence. The following makes this precise.

  • Let In be the meet of the n th tail of the sequence. That is,
  • Similarly, let Jn be the join of the n th tail of the sequence. That is,

Examples Edit

The following are several set convergence examples. They have been broken into sections with respect to the metric used to induce the topology on set X.

  • The Borel–Cantelli lemma is an example application of these constructs.
  • Consider the set X = <0,1>and the sequence of subsets:
  • lim sup Xn =
  • lim inf Xn = <>
  • lim sup Yn = lim inf Yn = lim Yn =
  • lim sup Zn = lim inf Zn = lim Zn =
  • Consider the set X = <50, 20, -100, -25, 0, 1>and the sequence of subsets:
  • lim sup Xn =
  • lim inf Xn = <>
  • Consider the sequence of subsets of rational numbers:
  • lim sup Xn =
  • lim inf Xn = <>
  • lim sup Yn = lim inf Yn = lim Yn =
  • lim sup Zn = lim inf Zn = lim Zn =
  • The Ω limit (i.e., limit set) of a solution to a dynamic system is the outer limit of solution trajectories of the system. [4] : 50–51 Because trajectories become closer and closer to this limit set, the tails of these trajectories converge to the limit set.
  • For example, an LTI system that is the cascade connection of several stable systems with an undamped second-order LTI system (i.e., zero damping ratio) will oscillate endlessly after being perturbed (e.g., an ideal bell after being struck). Hence, if the position and velocity of this system are plotted against each other, trajectories will approach a circle in the state space. This circle, which is the Ω limit set of the system, is the outer limit of solution trajectories of the system. The circle represents the locus of a trajectory corresponding to a pure sinusoidal tone output that is, the system output approaches/approximates a pure tone.

The above definitions are inadequate for many technical applications. In fact, the definitions above are specializations of the following definitions.

Definition for a set Edit

The limit inferior of a set XY is the infimum of all of the limit points of the set. That is,

Similarly, the limit superior of a set X is the supremum of all of the limit points of the set. That is,

Note that the set X needs to be defined as a subset of a partially ordered set Y that is also a topological space in order for these definitions to make sense. Moreover, it has to be a complete lattice so that the suprema and infima always exist. In that case every set has a limit superior and a limit inferior. Also note that the limit inferior and the limit superior of a set do not have to be elements of the set.

Definition for filter bases Edit

Take a topological space X and a filter base B in that space. The set of all cluster points for that filter base is given by

where B ¯ 0 >_<0>> is the closure of B 0 > . This is clearly a closed set and is similar to the set of limit points of a set. Assume that X is also a partially ordered set. The limit superior of the filter base B is defined as

when that supremum exists. When X has a total order, is a complete lattice and has the order topology,

Similarly, the limit inferior of the filter base B is defined as

when that infimum exists if X is totally ordered, is a complete lattice, and has the order topology, then

If the limit inferior and limit superior agree, then there must be exactly one cluster point and the limit of the filter base is equal to this unique cluster point.

Specialization for sequences and nets Edit

Therefore, the limit inferior and limit superior of the sequence are equal to the limit superior and limit inferior of C respectively.


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Population standard deviation of grades of eight students Edit

Suppose that the entire population of interest is eight students in a particular class. For a finite set of numbers, the population standard deviation is found by taking the square root of the average of the squared deviations of the values subtracted from their average value. The marks of a class of eight students (that is, a statistical population) are the following eight values:

These eight data points have the mean (average) of 5:

First, calculate the deviations of each data point from the mean, and square the result of each:

The variance is the mean of these values:

and the population standard deviation is equal to the square root of the variance:

This formula is valid only if the eight values with which we began form the complete population. If the values instead were a random sample drawn from some large parent population (for example, they were 8 students randomly and independently chosen from a class of 2 million), then one divides by 7 (which is n − 1) instead of 8 (which is n) in the denominator of the last formula, and the result is s = 32 / 7 ≈ 2.1. < extstyle s=>approx 2.1.> In that case, the result of the original formula would be called the sample standard deviation and denoted by s instead of σ . Dividing by n − 1 rather than by n gives an unbiased estimate of the variance of the larger parent population. This is known as Bessel's correction. [5] [6] Roughly, the reason for it is that the formula for the sample variance relies on computing differences of observations from the sample mean, and the sample mean itself was constructed to be as close as possible to the observations, so just dividing by n would underestimate the variability.

Standard deviation of average height for adult men Edit

If the population of interest is approximately normally distributed, the standard deviation provides information on the proportion of observations above or below certain values. For example, the average height for adult men in the United States is about 70 inches (177.8 cm), with a standard deviation of around 3 inches (7.62 cm). This means that most men (about 68%, assuming a normal distribution) have a height within 3 inches (7.62 cm) of the mean (67–73 inches (170.18–185.42 cm)) – one standard deviation – and almost all men (about 95%) have a height within 6 inches (15.24 cm) of the mean (64–76 inches (162.56–193.04 cm)) – two standard deviations. If the standard deviation were zero, then all men would be exactly 70 inches (177.8 cm) tall. If the standard deviation were 20 inches (50.8 cm), then men would have much more variable heights, with a typical range of about 50–90 inches (127–228.6 cm). Three standard deviations account for 99.7% of the sample population being studied, assuming the distribution is normal or bell-shaped (see the 68-95-99.7 rule, or the empirical rule, for more information).

Let μ be the expected value (the average) of random variable X with density f(x):

The standard deviation σ of X is defined as

Using words, the standard deviation is the square root of the variance of X.

The standard deviation of a probability distribution is the same as that of a random variable having that distribution.

Not all random variables have a standard deviation. If the distribution has fat tails going out to infinity, the standard deviation might not exist, because the integral might not converge. The normal distribution has tails going out to infinity, but its mean and standard deviation do exist, because the tails diminish quickly enough. The Pareto distribution with parameter α ∈ ( 1 , 2 ] has a mean, but not a standard deviation (loosely speaking, the standard deviation is infinite). The Cauchy distribution has neither a mean nor a standard deviation.

Discrete random variable Edit

In the case where X takes random values from a finite data set x1, x2, …, xN, with each value having the same probability, the standard deviation is

If, instead of having equal probabilities, the values have different probabilities, let x1 have probability p1, x2 have probability p2, …, xN have probability pN. In this case, the standard deviation will be

Continuous random variable Edit

and where the integrals are definite integrals taken for x ranging over the set of possible values of the random variable X.

In the case of a parametric family of distributions, the standard deviation can be expressed in terms of the parameters. For example, in the case of the log-normal distribution with parameters μ and σ 2 , the standard deviation is

One can find the standard deviation of an entire population in cases (such as standardized testing) where every member of a population is sampled. In cases where that cannot be done, the standard deviation σ is estimated by examining a random sample taken from the population and computing a statistic of the sample, which is used as an estimate of the population standard deviation. Such a statistic is called an estimator, and the estimator (or the value of the estimator, namely the estimate) is called a sample standard deviation, and is denoted by s (possibly with modifiers).

Unlike in the case of estimating the population mean, for which the sample mean is a simple estimator with many desirable properties (unbiased, efficient, maximum likelihood), there is no single estimator for the standard deviation with all these properties, and unbiased estimation of standard deviation is a very technically involved problem. Most often, the standard deviation is estimated using the corrected sample standard deviation (using N − 1), defined below, and this is often referred to as the "sample standard deviation", without qualifiers. However, other estimators are better in other respects: the uncorrected estimator (using N) yields lower mean squared error, while using N − 1.5 (for the normal distribution) almost completely eliminates bias.

Uncorrected sample standard deviation Edit

The formula for the population standard deviation (of a finite population) can be applied to the sample, using the size of the sample as the size of the population (though the actual population size from which the sample is drawn may be much larger). This estimator, denoted by sN, is known as the uncorrected sample standard deviation, or sometimes the standard deviation of the sample (considered as the entire population), and is defined as follows: [7]

This is a consistent estimator (it converges in probability to the population value as the number of samples goes to infinity), and is the maximum-likelihood estimate when the population is normally distributed. [ citation needed ] However, this is a biased estimator, as the estimates are generally too low. The bias decreases as sample size grows, dropping off as 1/N, and thus is most significant for small or moderate sample sizes for N > 75 the bias is below 1%. Thus for very large sample sizes, the uncorrected sample standard deviation is generally acceptable. This estimator also has a uniformly smaller mean squared error than the corrected sample standard deviation.

Corrected sample standard deviation Edit

If the biased sample variance (the second central moment of the sample, which is a downward-biased estimate of the population variance) is used to compute an estimate of the population's standard deviation, the result is

Here taking the square root introduces further downward bias, by Jensen's inequality, due to the square root's being a concave function. The bias in the variance is easily corrected, but the bias from the square root is more difficult to correct, and depends on the distribution in question.

An unbiased estimator for the variance is given by applying Bessel's correction, using N − 1 instead of N to yield the unbiased sample variance, denoted s 2 :

This estimator is unbiased if the variance exists and the sample values are drawn independently with replacement. N − 1 corresponds to the number of degrees of freedom in the vector of deviations from the mean, ( x 1 − x ¯ , … , x n − x ¯ ) . -<ar >,dots ,x_-<ar >).>

Taking square roots reintroduces bias (because the square root is a nonlinear function, which does not commute with the expectation), yielding the corrected sample standard deviation, denoted by s: [2]

As explained above, while s 2 is an unbiased estimator for the population variance, s is still a biased estimator for the population standard deviation, though markedly less biased than the uncorrected sample standard deviation. This estimator is commonly used and generally known simply as the "sample standard deviation". The bias may still be large for small samples (N less than 10). As sample size increases, the amount of bias decreases. We obtain more information and the difference between 1 N >> and 1 N − 1 >> becomes smaller.

Unbiased sample standard deviation Edit

For unbiased estimation of standard deviation, there is no formula that works across all distributions, unlike for mean and variance. Instead, s is used as a basis, and is scaled by a correction factor to produce an unbiased estimate. For the normal distribution, an unbiased estimator is given by s/c4, where the correction factor (which depends on N) is given in terms of the Gamma function, and equals:

This arises because the sampling distribution of the sample standard deviation follows a (scaled) chi distribution, and the correction factor is the mean of the chi distribution.

An approximation can be given by replacing N − 1 with N − 1.5, yielding:

The error in this approximation decays quadratically (as 1/N 2 ), and it is suited for all but the smallest samples or highest precision: for N = 3 the bias is equal to 1.3%, and for N = 9 the bias is already less than 0.1%.

For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of the approximation:

where γ2 denotes the population excess kurtosis. The excess kurtosis may be either known beforehand for certain distributions, or estimated from the data. [ citation needed ]

Confidence interval of a sampled standard deviation Edit

The standard deviation we obtain by sampling a distribution is itself not absolutely accurate, both for mathematical reasons (explained here by the confidence interval) and for practical reasons of measurement (measurement error). The mathematical effect can be described by the confidence interval or CI.

To show how a larger sample will make the confidence interval narrower, consider the following examples: A small population of N = 2 has only 1 degree of freedom for estimating the standard deviation. The result is that a 95% CI of the SD runs from 0.45 × SD to 31.9 × SD the factors here are as follows:

A larger population of N = 10 has 9 degrees of freedom for estimating the standard deviation. The same computations as above give us in this case a 95% CI running from 0.69 × SD to 1.83 × SD. So even with a sample population of 10, the actual SD can still be almost a factor 2 higher than the sampled SD. For a sample population N=100, this is down to 0.88 × SD to 1.16 × SD. To be more certain that the sampled SD is close to the actual SD we need to sample a large number of points.

These same formulae can be used to obtain confidence intervals on the variance of residuals from a least squares fit under standard normal theory, where k is now the number of degrees of freedom for error.

Bounds on standard deviation Edit

For a set of N > 4 data spanning a range of values R, an upper bound on the standard deviation s is given by s = 0.6R. [9] An estimate of the standard deviation for N > 100 data taken to be approximately normal follows from the heuristic that 95% of the area under the normal curve lies roughly two standard deviations to either side of the mean, so that, with 95% probability the total range of values R represents four standard deviations so that s ≈ R/4. This so-called range rule is useful in sample size estimation, as the range of possible values is easier to estimate than the standard deviation. Other divisors K(N) of the range such that s ≈ R/K(N) are available for other values of N and for non-normal distributions. [10]

The standard deviation is invariant under changes in location, and scales directly with the scale of the random variable. Thus, for a constant c and random variables X and Y:

The standard deviation of the sum of two random variables can be related to their individual standard deviations and the covariance between them:

The calculation of the sum of squared deviations can be related to moments calculated directly from the data. In the following formula, the letter E is interpreted to mean expected value, i.e., mean.

The sample standard deviation can be computed as:

For a finite population with equal probabilities at all points, we have

which means that the standard deviation is equal to the square root of the difference between the average of the squares of the values and the square of the average value.

See computational formula for the variance for proof, and for an analogous result for the sample standard deviation.

A large standard deviation indicates that the data points can spread far from the mean and a small standard deviation indicates that they are clustered closely around the mean.

For example, each of the three populations <0, 0, 14, 14>, <0, 6, 8, 14>and <6, 6, 8, 8>has a mean of 7. Their standard deviations are 7, 5, and 1, respectively. The third population has a much smaller standard deviation than the other two because its values are all close to 7. These standard deviations have the same units as the data points themselves. If, for instance, the data set <0, 6, 8, 14>represents the ages of a population of four siblings in years, the standard deviation is 5 years. As another example, the population <1000, 1006, 1008, 1014>may represent the distances traveled by four athletes, measured in meters. It has a mean of 1007 meters, and a standard deviation of 5 meters.

Standard deviation may serve as a measure of uncertainty. In physical science, for example, the reported standard deviation of a group of repeated measurements gives the precision of those measurements. When deciding whether measurements agree with a theoretical prediction, the standard deviation of those measurements is of crucial importance: if the mean of the measurements is too far away from the prediction (with the distance measured in standard deviations), then the theory being tested probably needs to be revised. This makes sense since they fall outside the range of values that could reasonably be expected to occur, if the prediction were correct and the standard deviation appropriately quantified. See prediction interval.

While the standard deviation does measure how far typical values tend to be from the mean, other measures are available. An example is the mean absolute deviation, which might be considered a more direct measure of average distance, compared to the root mean square distance inherent in the standard deviation.

Application examples Edit

The practical value of understanding the standard deviation of a set of values is in appreciating how much variation there is from the average (mean).

Experiment, industrial and hypothesis testing Edit

Standard deviation is often used to compare real-world data against a model to test the model. For example, in industrial applications the weight of products coming off a production line may need to comply with a legally required value. By weighing some fraction of the products an average weight can be found, which will always be slightly different from the long-term average. By using standard deviations, a minimum and maximum value can be calculated that the averaged weight will be within some very high percentage of the time (99.9% or more). If it falls outside the range then the production process may need to be corrected. Statistical tests such as these are particularly important when the testing is relatively expensive. For example, if the product needs to be opened and drained and weighed, or if the product was otherwise used up by the test.

In experimental science, a theoretical model of reality is used. Particle physics conventionally uses a standard of "5 sigma" for the declaration of a discovery. A five-sigma level translates to one chance in 3.5 million that a random fluctuation would yield the result. This level of certainty was required in order to assert that a particle consistent with the Higgs boson had been discovered in two independent experiments at CERN, [11] also leading to the declaration of the first observation of gravitational waves, [12] and confirmation of global warming. [13]

Weather Edit

As a simple example, consider the average daily maximum temperatures for two cities, one inland and one on the coast. It is helpful to understand that the range of daily maximum temperatures for cities near the coast is smaller than for cities inland. Thus, while these two cities may each have the same average maximum temperature, the standard deviation of the daily maximum temperature for the coastal city will be less than that of the inland city as, on any particular day, the actual maximum temperature is more likely to be farther from the average maximum temperature for the inland city than for the coastal one.

Finance Edit

In finance, standard deviation is often used as a measure of the risk associated with price-fluctuations of a given asset (stocks, bonds, property, etc.), or the risk of a portfolio of assets [14] (actively managed mutual funds, index mutual funds, or ETFs). Risk is an important factor in determining how to efficiently manage a portfolio of investments because it determines the variation in returns on the asset and/or portfolio and gives investors a mathematical basis for investment decisions (known as mean-variance optimization). The fundamental concept of risk is that as it increases, the expected return on an investment should increase as well, an increase known as the risk premium. In other words, investors should expect a higher return on an investment when that investment carries a higher level of risk or uncertainty. When evaluating investments, investors should estimate both the expected return and the uncertainty of future returns. Standard deviation provides a quantified estimate of the uncertainty of future returns.

For example, assume an investor had to choose between two stocks. Stock A over the past 20 years had an average return of 10 percent, with a standard deviation of 20 percentage points (pp) and Stock B, over the same period, had average returns of 12 percent but a higher standard deviation of 30 pp. On the basis of risk and return, an investor may decide that Stock A is the safer choice, because Stock B's additional two percentage points of return is not worth the additional 10 pp standard deviation (greater risk or uncertainty of the expected return). Stock B is likely to fall short of the initial investment (but also to exceed the initial investment) more often than Stock A under the same circumstances, and is estimated to return only two percent more on average. In this example, Stock A is expected to earn about 10 percent, plus or minus 20 pp (a range of 30 percent to −10 percent), about two-thirds of the future year returns. When considering more extreme possible returns or outcomes in future, an investor should expect results of as much as 10 percent plus or minus 60 pp, or a range from 70 percent to −50 percent, which includes outcomes for three standard deviations from the average return (about 99.7 percent of probable returns).

Calculating the average (or arithmetic mean) of the return of a security over a given period will generate the expected return of the asset. For each period, subtracting the expected return from the actual return results in the difference from the mean. Squaring the difference in each period and taking the average gives the overall variance of the return of the asset. The larger the variance, the greater risk the security carries. Finding the square root of this variance will give the standard deviation of the investment tool in question.

Financial time series are known to be non-stationary series, whereas the statistical calculations above, such as standard deviation, apply only to stationary series. To apply the above statistical tools to non-stationary series, the series first must be transformed to a stationary series, enabling use of statistical tools that now have a valid basis from which to work.

Geometric interpretation Edit

To gain some geometric insights and clarification, we will start with a population of three values, x1, x2, x3. This defines a point P = (x1, x2, x3) in R 3 . Consider the line L = <(r, r, r) : rR>. This is the "main diagonal" going through the origin. If our three given values were all equal, then the standard deviation would be zero and P would lie on L. So it is not unreasonable to assume that the standard deviation is related to the distance of P to L. That is indeed the case. To move orthogonally from L to the point P, one begins at the point:


Solving for a Variable on One Side Using Multiplication

Sometimes our variable is being multiplied to a number, in this case, we use the multiplicative inverse (which we learned about in our lessons on fractions) to isolate our variable. In all cases when we&rsquore solving for variables, it is important to remember that anything we do to one side of the equation, we must do to the other.

This video uses the word isolate.

As shown in the video, to isolate a variable when it&rsquos being multiplied, we multiply both sides of the equation with the multiplicative inverse of the number. Remember, the multiplicative inverse is the opposite fraction (ex: (3) and (frac<1><3>) ).

Additional Resources

Solve for the variable:

Solutions

In this example, we want to get the variable (< ext>) alone on one side of the equal sign in order to find out what it is equal to. (< ext>) is currently being multiplied by (5). We can remove the (5) by multiplying both sides by the multiplicative inverse of (5).

The multiplicative inverse of a number is the number that when multiplied to it, the product is (1).

We are looking for (5 imes ?> = 1)

The multiplicative inverse of (5) is (frac<1><5>), because (5left (frac<1> <5> ight )=1)

We multiply both sides of the equation by (frac<1><5>).

Since (frac<1><5>) multiplied to (5) equals (1), we are left with (1< ext>) on the left side.

(1< ext>) is the same as just (< ext>) since anything times (1) is itself.

On the right-hand side of the equation, (frac<1><5>) times (25) is the same as (frac<1><5>) times (frac<25><1>) since anything divided by (1) is still itself.

Then we multiply across the numerator and denominator when multiplying fractions.

In this example, we want to get the variable (< ext>) alone on one side of the equal sign in order to find out what it is equal to. (< ext>) is currently being multiplied by (4). We can remove the (4) by multiplying both sides by the multiplicative inverse of (4).

The multiplicative inverse of a number is the number that when multiplied to it, the product is (1).

The multiplicative inverse of (4) is (frac<1><4>), because (4left ( frac<1> <4> ight )=1).

We multiply both sides of the equation by (frac<1><4>).

Since (frac<1><4>) multiplied to (4) equals (1), we are left with (1< ext>) on the left side.

(1< ext>) is the same as just (< ext>) since anything times (1) is itself.

On the right-hand side of the equation, (frac<1><4>) times (-24) is the same as (-24) divided by (4) after multiplying across.


One-Sided Limits

Our topic of discussion in this section is one-sided limits, which builds upon the preceding lesson on continuity. Create a new worksheet called 03-One-Sided Limits. The basis of one-sided limits is that when a function jumps suddenly from one value to another, it often is not possible to describe the function's behavior with a single limit. What we can do, though, is to describe the function's behavior from the right and from the left using two limits. Consider the following graph, the code of which is provided:

1-2) Plot -x 2 +6 from 0 to 2, x-1 from 2 to 4
3-6) Create three closed points, one open
7) Combine the plots and points, then show the result with the given x and y boundaries

The above function has a discontinuity at x=2, and since the two pieces of the function approach different values:

You probably see where this is going. What we can say that the limit of f(x) as x approaches 2 from the left is 2, and the limit of f(x) as x approaches 2 from the right is 1. If you were to write this, it would look like:

The minus sign indicates "from the left", and the plus sign indicates "from the right". Since the limit of f(x) as x approaches 2 from the right is equal to f(2), f(x) is said to be continuous from the right at 2. The limit of f(x) as x approaches 2 from the left does not equal f(2), however, so f(x) is not continuous from the left at 2.

One-sided limits are usually fairly straightforward. However, be aware that when a function approaches a vertical asymptote, such as at x=0 in the following graph, you would describe the limit of the function as approaching -oo or oo, depending on the case. A vertical asymptote is an x-value of a function at which one or both sides approach infinity or negative infinity.

1) Plot 1/x from -6 to 6. randomize=False produces a more consistent result when this particular function is plotted.

Here, we would say that the limit of f(x) as x approaches zero from the left is negative infinity and that the limit of f(x) as x approaches zero from the right is infinity. The limit of f(x) as x approaches zero is undefined, since both sides approach different values. Visually,

, , and is undefined.

A Math Problem

Sue and Bob take turns rolling a 6-sided die. Once either person rolls a 6, the game is over. Sue rolls first. If she doesn’t roll a 6, Bob rolls the die if he doesn’t roll a 6, Sue rolls again. They continue taking turns until one of them rolls a 6.

Bob rolls a 6 before Sue.

What is the probability Bob rolled the 6 on his second turn?

The answer is not 5/36.

I love puzzles which are simple to state but have a fiendishly tricky or counterintuitive answer. I just threw up a page on the xkcd IRC wiki to hold some of the better ones I’ve found over the years. I’ll be adding more over the next few weeks as I remember or find good ones. Feel free to add some of your own!

Edit: Buttons and then Daniel Barkalow got the correct answer first. Here it is, rot13‘d. Check your answer against this before posting smugly or people (I) will tease you: gjb friragl svir bire gjryir avargl fvk, be nobhg gjragl-bar cbvag bar creprag.


Calculate the limit of an expression

A limit is a certain value to which a function approaches. Finding a limit usually means finding what value y is as x approaches a certain number. You would typical phrase it as something like "the limit of a function f(x) is 7 as x approaches infinity. For example, imagine a curve such that as x approaches infinity, that curve comes closer and closer to y=0 while never actually getting there. So, how do we algebraically find that limit? One way to find the limit is by the substitution method.

For example, the limit of the following graph is 0 as x approaches infinity, clearly seen as the graph approaches 0 like so:

Now, let's look at a few examples where we can find the limit of real functions:

Example A

Find the limit of (f(x) = 4x), as x approaches 3.

1) Replace x for 3.
2) Simplify.

(f(x) = 4x) becomes (f(3) = 4(3) = 12).

So, the limit of (f(x) = 4x) as x approaches 3 is 12.

In this case the solution was straightforward, because the function not only approaches 12 but goes right through it!

Example B: Find the limit:

Follow the same steps as above.

So, the limit of (x^2 + 5x - 3) as x approaches 1 is 3.

However, the substitution method will not always work. For Example C below, you must factor the numerator first BEFORE applying the substitution method.

Example C:

If we substitute 0 for x in Example C, we will create division by zero which DOES NOT EXIST or is UNDEFINED. That's is the reason factoring MUST be our first step in this sample. We have to clean it up a bit so there's no division by zero.

Factoring the numerator for x, which is common to both terms, gives us:

We cancelled a factor of x in the numerator and denominator, leaving us with a simple limit:

Now, we can substitute 0 for x to find the limit is -7:

Note: Even though we were able to simplify the function in Sample C by factoring, we can't pretend that it didn't happen. Remember that we were finding the limit as x approached 0, not trying to evaluate the function AT x=0. The function is still undefined at x=0. It does, however, have a limit. Only the simplified version has a solution at x=0. Only after factoring, in some cases, can we then apply substitution to find the limit.


How does Minitab calculate %tolerance when I enter a one-sided tolerance

  1. Minitab calculates a one-sided process variation by dividing the study variation statistic by 2.
  2. Minitab defines the one-sided tolerance as the absolute value of the difference between the single specification limit and the mean value of all measurements.
  3. Minitab calculates the %tolerance statistic by dividing the one-sided process variation by the one-sided tolerance.

TermDescription
Lthe single specification limit
the mean of all observations

If the mean of all observations is less than the lower specification limit, or greater than the upper specification limit, the measurements deviate strongly from their acceptable range, and % tolerance is not calculated.


Watch the video: Calculus 1: Lecture Continuity and One-Sided Limits (December 2021).