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3.6: Limit Superior and Limit Inferior of Functions

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We extend to functions and concepts of limit superior and limit inferior.

Definition 3.6.1

Let f:ER and let ˉx be a limit point of D. Recall that

B0(ˉx;δ)=B(ˉx;δ)B+(ˉx;δ)=(ˉxδ,ˉx)(ˉx,ˉx+δ).

The limit superior of the function f at ˉx is defnied by

lim supxˉxf(x)=infδ>0supxB0(ˉx;δ)Df(x).

Similarly, the limit inferior of the function f at ˉx is defineid by

lim infxˉxf(x)=supδ>0infxB0(ˉx;δ)Df(x).

Consider the extended real-valued function g:(0,)(,] defined by

g(δ)=supxB0(x;δ)Df(x)

It is clear that g is increasing and

lim supxˉxf(x)=infδ>0g(δ).

We say that the function f is locally bounded above around ˉx if there exists δ>0 and M>0 such that

f(x)M for all xB(ˉx;δ)D.

Clearly, if f is locally bounded above around ˉx, then lim supxˉxf(x) is a real number, while lim supxzf(x)= in the other case. Similar discussion applies for the limit inferior.

Theorem 3.6.1

Let f:DR and let ˉx be a limit point of D. Then =lim supxˉxf(x) if and only if the following two conditions hold:

  1. For every ε>0, there exists δ>0 such that

f(x)<+ε for all xB0(ˉx;δ)D;

  1. For every ε>0 and for every δ>0, there exists xδB0(ˉx;δ)D such that

ε<f(xδ)

Proof

Suppose =lim supxˉxf(x). Then

=infδ>0g(δ),

where g is defined in (3.10). For any ε>0, there exists δ>0 such that

g(δ)=supxB0(ˉx;δ)Df(x)<+ε.

Thus,

f(x)<+ε for all xB0(ˉx;δ)D,

which proves conditions (1). Next note that for any ε>0 and δ>0, we have

ε<g(δ)=supxB0(ˉx;δ)Df(x).

Thus, there exists xδB0(ˉx;δ)D with

ε<f(xδ).

This proves (2).

Let us now prove the converse. Suppose (1) and (2) are satisfied. Fix any ε>0 and let δ>0 satisfy (1). Then

g(δ)=supxB0(ˉx;δ)Df(x)+ε.

This implies

lim supxˉxf(x)=infδ>0g(δ)+ε.

Since ε is arbitrary, we get

lim supxˉxf(x).

Again, let ε>0. Given δ>0, let xδ be as in (2). Therefore,

ε<f(xδ)supxB0(ˉx;δ)Df(x)=g(δ).

This implies

εinfδ>0g(δ)=lim supxˉxf(x).

It follows that lim supxˉxf(x). Therefore, =lim supxˉxf(x).

Corollary 3.6.2

Suppose =lim supxˉxf(x). Then there exists a sequence {xk} in D such that {xk} converges to ˉx, xkˉx for every k, and

limkf(xk)=.

Moreover, if {yk} is a sequence in D that converges to ˉx, ykˉx for every k, and limkf(yk)=, then .

Proof

Let δk=min{δk,1k}. Then δkδk and limkδk=0. From (2) of Theorem 3.6.1, there exists xkB0(ˉx;δk)D such that

εk<f(xk).

Moreover, f(xk)<+εk by (3.11). Therefore, {xk} is a sequence that satisfies the conclusion of the corollary.

Now let {yk} be a sequence in D that converges to ˉx, ykˉx for every k, and limkf(yk)=. For any ε>0, let δ>0 be as in (1) of Theorem 3.6.1. Since ykB0(ˉx;δ)D when k is sufficiently large, we have

f(yk)<+ε

for such k. This implies +ε. It follows that .

Remark 3.6.3

Let f:DR and let ˉx be a limit point of D. Suppose lim supxˉxf(x) is a real number. Define

A={R:{xk}D,xkˉx for every k,xkˉx,f(xk)}.

Then the previous corollary shows that A and lim supxˉxf(x)=maxA.

Theorem 3.6.4

Let f:DR and let ˉx be a limit point of D. Then

lim supxˉxf(x)=

if and only if there exists a sequence {xk} in D such that {xk} converges to ˉx, xkˉx for every k, and limkf(xk)=.

Proof

Suppose lim supxˉxf(x)=. Then

infδ>0g(δ)=,

wehre g is the extended real-valued function defined in (3.10). Thus, g(δ)= for every δ>0. Given kN, for δk=1k, since

g(δk)=supxB0(ˉx;δk)Df(x)=,

there exists xkB0(ˉx;δk)D such that f(xk)>k. Therefore, limkf(xk)=.

Let us prove the converse. Since limkf(xk)=, for every MR, there exists KN such that

f(xk)M for every kK.

For any δ>0, we have

xkB0(ˉx;δ)D.

This implies g(δ)=, and hence lim supxˉxf(x)=.

Theorem 3.6.5

Let f:DR and let ˉx be a limit point of D. Then

lim supxˉxf(x)=

if and only if for any sequence {xk} in D such that {xk} converges to ˉx, xkˉx for every k, it follows that limkf(xk)=. The latter is equivalent to limxˉxf(x)=.

Following the same arguments, we can prove similar results for inferior limits of functions.

Proof

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Theorem 3.6.6

Let f:DR and let ˉx be a limit point of D. Then =lim infxˉxf(x) if and only if the following two conditions hold:

  1. For every ε>0, there exists δ>0 such that

ε<f(x) for all xB0(ˉx;δ)D;

  1. For every ε>0 and for every δ>0, there exists xB0(ˉx;δ)D such that

f(x)<+ε.

Proof

Add proof here and it will automatically be hidden

Theorem 3.6.7

Suppose =lim infxˉxf(x). Then there exists a sequence {xk} in D such that xk converges to ˉx, xkˉx for every k, and

limkf(xk)=.

Moreover, if {yk} is a sequence in D that converges to ˉx, ykˉx for every k, and limkf(yk)=, then .

Proof

Add proof here and it will automatically be hidden

Remark 3.6.8

Let f:DR and let ˉx be a limit point of D. Suppose lim infxˉxf(x) is a real number. Define

B={R:{xk}D,xkˉx for every k,xkˉx,f(xk)}.

Then B and lim infxˉxf(x)=minB.

Theorem 3.6.9

Let f:DR and let ˉx be a limit point of D. Then

lim infxˉxf(x)=

if and only if there exists a sequence {xk} in D such that {xk} converges to ˉx, xkˉx for every k, and limkf(xk)=.

Proof

Add proof here and it will automatically be hidden

Theorem 3.6.10

Let f:DR and let ˉx be a limit point of D. Then

lim infxˉxf(x)=

if and only if there exists a sequence {xk} in D such that {xk} converges to ˉx, xkˉx for every k, it follows limkf(xk)=. The latter is equivalent to limxˉxf(x)=.

Proof

Add proof here and it will automatically be hidden

Theorem 3.6.11

Let f:DR and let ˉx be a limit point of D. Then

limxˉxf(x)=.

if and only if

lim supxˉxf(x)=lim infxˉxf(x)=.

Proof

Suppose

limxˉxf(x)=.

Then for every ε>0, there exists δ>0 such that

ε<f(x)<+ε for all xB0(ˉx;δ)D.

Since this also holds for every 0<δ<δ, we get

ε<g(δ)+ε.

It follows that

εinfδ>0g(δ)+ε.

Therefore, lim supxˉxf(x)= since ε is arbitrary. The proof for the limit inferior is similar. The converse follows directly from (1) of Theorem 3.6.1 and Theorem 3.6.6.

Exercise 3.6.1

Let DR, f:DR, and ˉx be a limit point of D. Prove that lim infxˉxf(x)lim supxˉxf(x).

Answer

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Exercise 3.6.2

Find each of the following limits:

  1. lim supx0sin(1x).
  2. lim infx0sin(1x).
  3. lim supx0cosxx.
  4. lim infx0cosxx.
Answer

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This page titled 3.6: Limit Superior and Limit Inferior of Functions is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Lafferriere, Lafferriere, and Nguyen (PDXOpen: Open Educational Resources) .

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