What Is L'Hôpital's Rule and When Do You Use It?

L'Hôpital's rule evaluates certain limits by differentiating the numerator and denominator separately. In IB Mathematics: Analysis and Approaches HL, use it only when direct substitution produces the indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

How the Rule Works

Begin with direct substitution into

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

If this gives one of the two valid indeterminate forms, and the functions satisfy the necessary differentiability conditions, then

$$\lim_{x \to a}\frac{f(x)}{g(x)}=\lim_{x \to a}\frac{f'(x)}{g'(x)},$$

provided the new limit exists or diverges appropriately. Differentiate the numerator and denominator independently; do not use the quotient rule.

These conditions matter because the rule compares the rates at which numerator and denominator approach zero or grow without bound. It does not say that every quotient may be differentiated. Before applying it, confirm that both functions are differentiable near the limiting value and that the denominator's derivative is not zero throughout the relevant punctured interval. For limits as $x$ approaches infinity, the same test and process apply without change.

The symbols $\frac{0}{0}$ and $\frac{\infty}{\infty}$ do not equal numbers. They show that substitution has not determined the limit.

Use this exam sequence: substitute directly; state the resulting form; apply the rule only if it is valid; simplify the new expression; and evaluate the final limit. After every application, check the form again. If the new limit is determinate, stop; repeated differentiation is not automatic.

For example,

$$\lim_{x \to 0}\frac{e^x-1}{x} =\lim_{x \to 0}\frac{e^x}{1}=1.$$

The rule may be applied repeatedly:

$$\lim_{x \to 0}\frac{1-\cos x}{x^2} =\lim_{x \to 0}\frac{\sin x}{2x} =\lim_{x \to 0}\frac{\cos x}{2} =\frac{1}{2}.$$

It can also establish divergence. For example, $\lim_{x \to \infty}\frac{x^2}{x}$ has form , and differentiation gives .

A common misconception is differentiating the whole quotient. Use $\frac{f'(x)}{g'(x)}$, not the quotient rule. Factorization or rationalization may sometimes be shorter.

IB Exam Technique

This is AA HL-only content, sub-topic AHL 5.13: Limits and L'Hôpital's rule, examinable on Papers 1, 2, and 3. Show substitution, identify the indeterminate form, and differentiate separately to secure method marks.