Practice Squeeze Theorem in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

If $g(x) \leq f(x) \leq h(x)$ near $x = a$ , and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$ , then $\lim_{x \to a} f(x) = L$ .

If $f$ is squeezed between two functions that both approach the same value $L$ , then $f$ has no choice—it must also approach $L$ . Like being caught between two walls closing in to the same point.

Showing a random 20 of 50 problems.

Example 1

medium

Find

\lim_{x\to 0} \dfrac{\sin(5x)}{5x}

Example 2

hard

Find

\lim_{x\to 0} \sin(x)\cos(1/x)

Example 3

hard

Find

\lim_{x\to 0^+} x\ln x

using a squeeze idea (granting

\lim_{x\to 0^+}\dfrac{\ln x}{1/\sqrt x}=0

Example 4

medium

Evaluate

\lim_{x\to 0} x^4 \cos(1/x)

Example 5

easy

Use the squeeze theorem to find

\displaystyle\lim_{x \to 0} x^2 \sin\!\left(\frac{1}{x}\right)

Example 6

medium

Use

|1-\cos x|\le \dfrac{x^2}{2}

to find

\lim_{x\to 0} \dfrac{1-\cos x}{x^2}

is bounded above by what?

Example 7

challenge

Show that if

|f(x)-L|\le M|x-a|^p

for some

M>0, p>0

and all

x

near

a

, then

\lim_{x\to a} f(x)=L

Example 8

medium

Evaluate

\lim_{n\to\infty} \dfrac{(-1)^n}{n}

Example 9

challenge

Given

|f(x) - 3| \leq |x - 2|

for all

x

, prove

\lim_{x\to2} f(x) = 3

Example 10

easy

Find

\displaystyle\lim_{x \to 0} x\cos\!\left(\frac{1}{x}\right)

Example 11

hard

Find

\lim_{n\to\infty} \dfrac{1+2+\cdots+\lfloor n\sin n\rfloor}{n^2}

bounded in absolute value.

Example 12

medium

Show that

\displaystyle\lim_{n\to\infty} \frac{\sin n}{n} = 0

Example 13

medium

Evaluate

\lim_{x\to0} x^2 \lfloor 1/x \rfloor

where

\lfloor\cdot\rfloor

is the floor.

Example 14

easy

Given

-x^2\le f(x)\le x^2

near

0

, find

\lim_{x\to 0} f(x)

Example 15

easy

Evaluate

\lim_{x\to0} x^2 \sin\!\left(\frac{1}{x}\right)

Example 16

medium

Find

\lim_{x\to\infty} \frac{2 + \cos x}{x}

Example 17

medium

Given

\cos x \leq \frac{\sin x}{x} \leq 1

near 0, find

\lim_{x\to0} \frac{\sin x}{x}

Example 18

challenge

Prove

\lim_{x\to0} x \lfloor 1/x \rfloor = 1

does not follow from naive bounds; find the correct limit.

Example 19

challenge

Use the squeeze theorem to find

\lim_{n\to\infty} \sqrt[n]{2^n + 3^n}

Example 20

medium

Find

\lim_{x\to0} \frac{1 - \cos x}{x}

using

0 \leq 1 - \cos x \leq \frac{x^2}{2}

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Related Concepts

Limit Types of Continuity and Discontinuity