Practice L'Hopital's Rule in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick 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.'
Example 1
easyFind \displaystyle\lim_{x \to 0} \frac{\sin x}{x} using L'Hôpital's rule.
Example 2
hardFind \displaystyle\lim_{x \to \infty} x e^{-x}.
Example 3
easyFind \displaystyle\lim_{x \to 1} \frac{x^2 - 1}{x - 1} using L'Hôpital's rule.
Example 4
mediumFind \displaystyle\lim_{x \to \infty} \frac{\ln x}{x}.