Practice L'Hopital's Rule 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 lim⁡x→af(x)g(x)\lim_{x \to a} \frac{f(x)}{g(x)} is an indeterminate form 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}, then lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} provided the right-hand limit exists (or is ±∞\pm\infty).

When both numerator and denominator go to zero (or both to infinity), the limit depends on which one gets there faster. Taking derivatives measures the rates at which they approach 0 or ∞\infty, so the ratio of derivatives captures this 'race.'

Showing a random 20 of 50 problems.

Example 1

easy
Evaluate lim⁡x→0tan⁡xx\lim_{x\to0} \frac{\tan x}{x}.

Example 2

hard
Evaluate lim⁡x→0arcsin⁡x−xx3\displaystyle\lim_{x\to 0}\frac{\arcsin x - x}{x^3}.

Example 3

hard
Evaluate lim⁡x→0+(1x−1ex−1)\displaystyle\lim_{x\to 0^+}\left(\dfrac{1}{x} - \dfrac{1}{e^x - 1}\right).

Example 4

easy
Is L'Hopital applicable to lim⁡x→1x2+1x+1\displaystyle\lim_{x\to 1}\frac{x^2+1}{x+1}?

Example 5

challenge
Evaluate lim⁡x→0+xx−1xln⁡x\displaystyle\lim_{x\to 0^+}\frac{x^x - 1}{x \ln x}.

Example 6

medium
Evaluate lim⁡x→∞(1+1x)x\lim_{x\to\infty} \left(1 + \frac{1}{x}\right)^x.

Example 7

challenge
Evaluate lim⁡x→0tan⁡x−xx3\lim_{x\to0} \frac{\tan x - x}{x^3}.

Example 8

medium
Evaluate lim⁡x→01−cos⁡xx2\displaystyle\lim_{x\to 0}\frac{1-\cos x}{x^2}.

Example 9

medium
Find lim⁡x→∞ln⁡xx\displaystyle\lim_{x \to \infty} \frac{\ln x}{x}.

Example 10

medium
Evaluate lim⁡x→∞x3ex\displaystyle\lim_{x\to\infty}\frac{x^3}{e^{x}}.

Example 11

easy
Evaluate lim⁡x→∞3x+52x−1\displaystyle\lim_{x\to\infty}\frac{3x+5}{2x-1}.

Example 12

hard
Evaluate lim⁡x→∞(x2+x−x)\displaystyle\lim_{x\to\infty}\left(\sqrt{x^2+x} - x\right).

Example 13

easy
Evaluate lim⁡x→0sin⁡xx\lim_{x\to0} \frac{\sin x}{x} using L'Hopital.

Example 14

easy
Evaluate lim⁡x→0sin⁡5xsin⁡2x\displaystyle\lim_{x\to 0}\frac{\sin 5x}{\sin 2x}.

Example 15

easy
Evaluate lim⁡x→3x2−9x−3\displaystyle\lim_{x\to 3}\frac{x^2-9}{x-3}.

Example 16

easy
Evaluate lim⁡x→0ex−1x\lim_{x\to0} \frac{e^x - 1}{x}.

Example 17

medium
Evaluate lim⁡x→0+(sin⁡x)x\displaystyle\lim_{x\to 0^+}(\sin x)^x.

Example 18

easy
Find lim⁡x→0sin⁡xx\displaystyle\lim_{x \to 0} \frac{\sin x}{x} using L'Hôpital's rule.

Example 19

medium
Evaluate lim⁡x→1ln⁡xx−1\displaystyle\lim_{x\to 1}\frac{\ln x}{x-1}.

Example 20

hard
Evaluate lim⁡x→0+(cos⁡x)1/x2\displaystyle\lim_{x\to 0^+}(\cos x)^{1/x^2}.
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