L'Hopital's Rule Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of L'Hopital's Rule.

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

Concept 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.'

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: For an indeterminate $\frac00$ or $\frac\infty\infty$ limit, replace it with the limit of the ratio of derivatives.

Common stuck point: The procedure for l'hopital's rule is the easy part; the trap is applying it to a non-indeterminate form. Asking "Does plugging in give $\frac00$ or $\frac\infty\infty$ in a quotient — and only then?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does plugging in give $\frac00$ or $\frac\infty\infty$ in a quotient — and only then?

Worked Examples

Example 1

easy

Find

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

using L'Hôpital's rule.

Answer

1

First step

Direct substitution:

\frac{\sin 0}{0} = \frac{0}{0}

— indeterminate form.

Full solution

2
Apply L'Hôpital: differentiate numerator and denominator separately.
3
$\lim_{x\to 0}\frac{\sin x}{x} = \lim_{x\to 0}\frac{\cos x}{1} = \frac{\cos 0}{1} = 1$ .

L'Hôpital's rule applies when we have a

\frac{0}{0}

form. Differentiate top and bottom independently (not using the quotient rule), then evaluate the limit.

Example 2

hard

Find

\displaystyle\lim_{x \to \infty} x e^{-x}

Example 3

easy

Evaluate

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

Example 4

medium

Evaluate

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

Example 5

medium

Evaluate

\displaystyle\lim_{x\to 0^+}\sqrt{x}\,\ln x

Example 6

hard

Evaluate

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

Example 7

hard

Evaluate

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

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 1} \frac{x^2 - 1}{x - 1}

using L'Hôpital's rule.

Example 2

medium

Find

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

Example 3

easy

Evaluate

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

using L'Hopital.

Example 4

easy

Evaluate

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

Example 5

easy

Evaluate

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

Example 6

easy

Evaluate

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

Example 7

easy

Should you apply L'Hopital to

\lim_{x\to0} \frac{\sin x}{x + 1}

Example 8

easy

Evaluate

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

Example 9

easy

Evaluate

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

Example 10

easy

Evaluate

\lim_{x\to2} \frac{x^2 - 4}{x - 2}

using L'Hopital.

Example 11

medium

Evaluate

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

Example 12

medium

Evaluate

\lim_{x\to0^+} x \ln x

Example 13

medium

Evaluate

\lim_{x\to\infty} \frac{x^2}{e^x}

Example 14

medium

Evaluate

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

Example 15

medium

Evaluate

\lim_{x\to0} \frac{e^x - 1 - x}{x^2}

Example 16

medium

Evaluate

\lim_{x\to0^+} \left(\frac{1}{x} - \frac{1}{\sin x}\right)

Example 17

medium

Evaluate

\lim_{x\to0} \frac{\ln(1 + x)}{x}

Example 18

medium

Evaluate

\lim_{x\to0^+} x^x

Example 19

medium

Evaluate

\lim_{x\to1} \frac{x^{10} - 1}{x - 1}

Example 20

challenge

Evaluate

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

Example 21

challenge

Evaluate

\lim_{x\to\infty} x\left(e^{1/x} - 1\right)

Example 22

challenge

Evaluate

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

Example 23

easy

Evaluate

\displaystyle\lim_{x\to 0}\frac{\sin 3x}{x}

Example 24

easy

Evaluate

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

Example 25

easy

Evaluate

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

Example 26

easy

Evaluate

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

Example 27

medium

Evaluate

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

Example 28

medium

Evaluate

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

Example 29

medium

Evaluate

\displaystyle\lim_{x\to 0}\frac{\ln(1+2x)}{x}

Example 30

medium

Evaluate

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

Example 31

medium

Evaluate

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

Example 32

medium

Evaluate

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

Example 33

medium

Evaluate

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

Example 34

medium

Evaluate

\displaystyle\lim_{x\to 0}\frac{e^x - e^{-x}}{\sin x}

Example 35

hard

Evaluate

\displaystyle\lim_{x\to\infty}\left(1+\dfrac{3}{x}\right)^{x}

Example 36

hard

Evaluate

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

Example 37

hard

Evaluate

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

Example 38

hard

Evaluate

\displaystyle\lim_{x\to 0}\frac{e^x - 1 - x - x^2/2}{x^3}

Example 39

hard

Evaluate

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

Example 40

challenge

Evaluate

\displaystyle\lim_{x\to 0}\frac{\sin(\sin x) - x}{x^3}

Example 41

challenge

Evaluate

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

← Back to L'Hopital's Rule Formula Explained Practice Mode

Related Concepts

Limit Derivative Infinity Taylor Series

Background Knowledge

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

limitderivativeinfinity