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)≤f(x)≤h(x)g(x) \leq f(x) \leq h(x) near x=ax = a, and lim⁡x→ag(x)=lim⁡x→ah(x)=L\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L, then lim⁡x→af(x)=L\lim_{x \to a} f(x) = L.

If ff is squeezed between two functions that both approach the same value LL, then ff has no choice—it must also approach LL. 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→0sin⁡(5x)5x\lim_{x\to 0} \dfrac{\sin(5x)}{5x}.

Example 2

hard
Find lim⁡x→0sin⁡(x)cos⁡(1/x)\lim_{x\to 0} \sin(x)\cos(1/x).

Example 3

hard
Find lim⁡x→0+xln⁡x\lim_{x\to 0^+} x\ln x using a squeeze idea (granting lim⁡x→0+ln⁡x1/x=0\lim_{x\to 0^+}\dfrac{\ln x}{1/\sqrt x}=0).

Example 4

medium
Evaluate lim⁡x→0x4cos⁡(1/x)\lim_{x\to 0} x^4 \cos(1/x).

Example 5

easy
Use the squeeze theorem to find lim⁡x→0x2sin⁡ ⁣(1x)\displaystyle\lim_{x \to 0} x^2 \sin\!\left(\frac{1}{x}\right).

Example 6

medium
Use ∣1−cos⁡x∣≤x22|1-\cos x|\le \dfrac{x^2}{2} to find lim⁡x→01−cos⁡xx2\lim_{x\to 0} \dfrac{1-\cos x}{x^2} is bounded above by what?

Example 7

challenge
Show that if ∣f(x)−L∣≤M∣x−a∣p|f(x)-L|\le M|x-a|^p for some M>0,p>0M>0, p>0 and all xx near aa, then lim⁡x→af(x)=L\lim_{x\to a} f(x)=L.

Example 8

medium
Evaluate lim⁡n→∞(−1)nn\lim_{n\to\infty} \dfrac{(-1)^n}{n}.

Example 9

challenge
Given ∣f(x)−3∣≤∣x−2∣|f(x) - 3| \leq |x - 2| for all xx, prove lim⁡x→2f(x)=3\lim_{x\to2} f(x) = 3.

Example 10

easy
Find lim⁡x→0xcos⁡ ⁣(1x)\displaystyle\lim_{x \to 0} x\cos\!\left(\frac{1}{x}\right).

Example 11

hard
Find lim⁡n→∞1+2+⋯+⌊nsin⁡n⌋n2\lim_{n\to\infty} \dfrac{1+2+\cdots+\lfloor n\sin n\rfloor}{n^2} bounded in absolute value.

Example 12

medium
Show that lim⁡n→∞sin⁡nn=0\displaystyle\lim_{n\to\infty} \frac{\sin n}{n} = 0.

Example 13

medium
Evaluate lim⁡x→0x2⌊1/x⌋\lim_{x\to0} x^2 \lfloor 1/x \rfloor where ⌊⋅⌋\lfloor\cdot\rfloor is the floor.

Example 14

easy
Given −x2≤f(x)≤x2-x^2\le f(x)\le x^2 near 00, find lim⁡x→0f(x)\lim_{x\to 0} f(x).

Example 15

easy
Evaluate lim⁡x→0x2sin⁡ ⁣(1x)\lim_{x\to0} x^2 \sin\!\left(\frac{1}{x}\right).

Example 16

medium
Find lim⁡x→∞2+cos⁡xx\lim_{x\to\infty} \frac{2 + \cos x}{x}.

Example 17

medium
Given cos⁡x≤sin⁡xx≤1\cos x \leq \frac{\sin x}{x} \leq 1 near 0, find lim⁡x→0sin⁡xx\lim_{x\to0} \frac{\sin x}{x}.

Example 18

challenge
Prove lim⁡x→0x⌊1/x⌋=1\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→∞2n+3nn\lim_{n\to\infty} \sqrt[n]{2^n + 3^n}.

Example 20

medium
Find lim⁡x→01−cos⁡xx\lim_{x\to0} \frac{1 - \cos x}{x} using 0≤1−cos⁡x≤x220 \leq 1 - \cos x \leq \frac{x^2}{2}.
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