Squeeze Theorem Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Squeeze Theorem.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: If $f$ is sandwiched between $g$ and $h$ that both approach $L$ , then $f$ must approach $L$ too.

Common stuck point: The procedure for squeeze theorem is the easy part; the trap is choosing bounds with unequal limits. Asking "Can I bound this function between two functions that approach the SAME limit at the point?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I bound this function between two functions that approach the SAME limit at the point?

Worked Examples

Example 1

easy

Use the squeeze theorem to find

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

Answer

0

First step

Since

-1 \leq \sin\!\left(\frac{1}{x}\right) \leq 1

for all

x \neq 0

Full solution

2
Multiply by $x^2 \geq 0$ : $-x^2 \leq x^2\sin\!\left(\frac{1}{x}\right) \leq x^2$ .
3
$\lim_{x\to 0}(-x^2) = 0$ and $\lim_{x\to 0}(x^2) = 0$ .
4
By the squeeze theorem: $\lim_{x\to 0} x^2\sin\!\left(\frac{1}{x}\right) = 0$ .

The function

\sin(1/x)

oscillates wildly near 0, so its limit doesn't exist alone. Multiplying by

x^2

traps it between

-x^2

and

x^2

, both going to 0, so the product must also go to 0.

Example 2

hard

Use the squeeze theorem to prove

\displaystyle\lim_{x \to 0} \frac{\sin x}{x} = 1

Example 3

medium

Use the squeeze theorem to evaluate

\lim_{x\to 0} \dfrac{x^2 \sin(1/x^2)}{x}

Example 4

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 5

hard

Find

\lim_{n\to\infty} \dfrac{n!}{n^n}

using the squeeze theorem.

Example 6

hard

Use the squeeze theorem to show

\lim_{x\to\infty}\dfrac{\sin x + \cos x}{\sqrt{x}}=0

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find

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

Example 2

medium

Show that

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

Example 3

easy

Given

-1 \leq f(x) \leq 1

and you multiply by

x^2

, what is

\lim_{x\to0} x^2 f(x)

Example 4

easy

Evaluate

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

Example 5

easy

g(x) \leq f(x) \leq h(x)

near

a

with

\lim g = \lim h = 5

, what is

\lim f

Example 6

easy

Evaluate

\lim_{x\to0} x \cos\!\left(\frac{1}{x}\right)

Example 7

easy

What bounds would you use to squeeze

x^4 \sin(1/x^2)

near 0?

Example 8

easy

Use boundedness:

\lim_{x\to\infty} \frac{\sin x}{x}

Example 9

easy

3 - x^2 \leq f(x) \leq 3 + x^2

, find

\lim_{x\to0} f(x)

Example 10

easy

True or false: the squeeze theorem applies if

\lim g = 2

and

\lim h = 4

Example 11

medium

Evaluate

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

Example 12

medium

Given

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

near 0, find

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

Example 13

medium

Evaluate

\lim_{x\to0^+} \sqrt{x}\, e^{\sin(1/x)}

Example 14

medium

Find

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

Example 15

medium

Evaluate

\lim_{n\to\infty} \frac{\sin(n)}{n^2}

for integer

n

Example 16

medium

Show

\lim_{x\to0} f(x) = 0

given

|f(x)| \leq 5x^2

Example 17

medium

Evaluate

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

where

\lfloor\cdot\rfloor

is the floor.

Example 18

medium

Find

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

using

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

Example 19

challenge

Prove

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

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

Example 20

medium

Evaluate

\lim_{x\to0} \frac{\sin(x^2)}{x}

using squeeze ideas.

Example 21

challenge

Use the squeeze theorem to find

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

Example 22

challenge

Given

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

for all

x

, prove

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

Example 23

easy

Given

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

near

0

, find

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

Example 24

easy

Evaluate

\lim_{x\to 0} x^3 \sin(1/x)

Example 25

easy

Find

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

Example 26

easy

Bound for the squeeze: write inequalities for

x^2 \cos(1/x)

valid near

0

Example 27

medium

Evaluate

\lim_{x\to 0} x^2 e^{\sin(1/x)}

Example 28

medium

Find

\lim_{x\to\infty} \dfrac{5+3\sin x}{x^2}

Example 29

medium

For

0<x<\tfrac{\pi}{2}

we have

\sin x<x

. Use the squeeze theorem (with

0\le \sin x\le x

) to find

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

Example 30

medium

Find

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

Example 31

medium

Evaluate

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

Example 32

hard

Find

\lim_{n\to\infty} \sqrt[n]{1^n+2^n+3^n}

Example 33

hard

Find

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

Example 34

hard

Find

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

bounded in absolute value.

Example 35

hard

Find

\lim_{n\to\infty} \sqrt[n]{n}

using the squeeze theorem (granting

\sqrt[n]{n}\ge 1

and

\sqrt[n]{n}\le 1+\sqrt{2/n}

for

n\ge 2

Example 36

challenge

Find

\lim_{n\to\infty} \dfrac{1}{n}\sum_{k=1}^{n}\dfrac{1}{\sqrt{n^2+k}}

using a squeeze.

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Related Concepts

Limit Types of Continuity and Discontinuity

Background Knowledge

These ideas may be useful before you work through the harder examples.

limit