How Do You Evaluate a Limit of the Indeterminate Form \(0/0\) or \(\infty/\infty\)?

First substitute the limiting value to confirm that the quotient has the indeterminate form $0/0$ or $\infty/\infty$. Then apply L’Hôpital’s rule by differentiating the numerator and denominator separately, and evaluate the new limit.

The Method

For suitable differentiable functions $f$ and $g$,

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

provided the original quotient produces $0/0$ or $\infty/\infty$, and the new limit exists or diverges to infinity. This is AHL 5.13: Limits and L’Hôpital’s rule in IB Mathematics: Analysis and Approaches HL.

The rule does not mean that $0/0$ has a numerical value. An indeterminate form means that direct substitution alone does not determine the limit. Before differentiating, check whether algebraic factorization, rationalization, or a known limit gives a shorter solution. L’Hôpital’s rule is available only after the required quotient form has been established.

For example,

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

Here, direct substitution first gives $0/0$. The trigonometric derivative assumes that $x$ is measured in radians.

Repeated application may be necessary:

$$\lim_{x \to \infty}\frac{3x^2+1}{2x^2-5} =\lim_{x \to \infty}\frac{6x}{4x} =\lim_{x \to \infty}\frac{6}{4} =\frac{3}{2}.$$

Another useful check is to simplify after each differentiation. In the second example, $6x/(4x)$ is already $3/2$ for $x\neq0$, so the second displayed differentiation is valid but unnecessary. Either route reaches the same limit, but concise, justified working reduces algebraic errors and communicates the method clearly.

A common misconception is to differentiate the entire quotient using the quotient rule. L’Hôpital’s rule requires (f'(x)/g'(x)), not the derivative of (f(x)/g(x)).

Exam Technique

On an AA HL exam, show the substitution that establishes the indeterminate form before applying the rule. For a calculate or find command, an examiner expects the relevant stages, including each repeated application. This is an analytical method suitable for Paper 1, where calculators are not permitted, and it may also be used on Paper 2 when appropriate.