Practice L'Hopital's Rule in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

\lim_{x \to a} \frac{f(x)}{g(x)}

is an indeterminate form

\frac{0}{0}

\frac{\infty}{\infty}

, then

\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\to0} \frac{\tan x}{x}

Example 2

hard

Evaluate

\displaystyle\lim_{x\to 0}\frac{\arcsin x - x}{x^3}

Example 3

hard

Evaluate

\displaystyle\lim_{x\to 0^+}\left(\dfrac{1}{x} - \dfrac{1}{e^x - 1}\right)

Example 4

easy

Is L'Hopital applicable to

\displaystyle\lim_{x\to 1}\frac{x^2+1}{x+1}

Example 5

challenge

Evaluate

\displaystyle\lim_{x\to 0^+}\frac{x^x - 1}{x \ln x}

Example 6

medium

Evaluate

\lim_{x\to\infty} \left(1 + \frac{1}{x}\right)^x

Example 7

challenge

Evaluate

\lim_{x\to0} \frac{\tan x - x}{x^3}

Example 8

medium

Evaluate

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

Example 9

medium

Find

\displaystyle\lim_{x \to \infty} \frac{\ln x}{x}

Example 10

medium

Evaluate

\displaystyle\lim_{x\to\infty}\frac{x^3}{e^{x}}

Example 11

easy

Evaluate

\displaystyle\lim_{x\to\infty}\frac{3x+5}{2x-1}

Example 12

hard

Evaluate

\displaystyle\lim_{x\to\infty}\left(\sqrt{x^2+x} - x\right)

Example 13

easy

Evaluate

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

using L'Hopital.

Example 14

easy

Evaluate

\displaystyle\lim_{x\to 0}\frac{\sin 5x}{\sin 2x}

Example 15

easy

Evaluate

\displaystyle\lim_{x\to 3}\frac{x^2-9}{x-3}

Example 16

easy

Evaluate

\lim_{x\to0} \frac{e^x - 1}{x}

Example 17

medium

Evaluate

\displaystyle\lim_{x\to 0^+}(\sin x)^x

Example 18

easy

Find

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

using L'Hôpital's rule.

Example 19

medium

Evaluate

\displaystyle\lim_{x\to 1}\frac{\ln x}{x-1}

Example 20

hard

Evaluate

\displaystyle\lim_{x\to 0^+}(\cos x)^{1/x^2}

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Related Concepts

Limit Derivative Infinity Taylor Series