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

If \lim_{x \to a} \frac{f(x)}{g(x)} is an indeterminate form \frac{0}{0} or \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: L'Hopital's rule only applies to indeterminate forms \frac{0}{0} or \frac{\infty}{\infty}. It replaces the limit of a ratio with the limit of the ratio of derivatives—you differentiate top and bottom separately (NOT using the quotient rule).

Common stuck point: You may need to apply L'Hopital's rule multiple times if the result is still indeterminate. For other indeterminate forms (0 \cdot \infty, 1^\infty, 0^0, \infty - \infty, \infty^0), rewrite first to get \frac{0}{0} or \frac{\infty}{\infty}.

Sense of Study hint: Before applying the rule, verify the form is 0/0 or infinity/infinity by substituting the limit value into top and bottom separately.

Worked Examples

Example 1

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

Solution

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

Answer

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}.

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}.
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Background Knowledge

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

limitderivativeinfinity