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Indeterminate Forms and L'Hôpital's Rule

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This concept page covers indeterminate forms and how L’Hôpital’s Rule helps evaluate them.

Indeterminate Forms

An indeterminate form is a limit expression that doesn’t immediately reveal the answer. The form doesn’t determine the limit - further analysis is needed.

The Seven Indeterminate Forms

Form Example
$\frac{0}{0}$ $\lim_{x \to 0} \frac{\sin x}{x}$
$\frac{\infty}{\infty}$ $\lim_{x \to \infty} \frac{e^x}{x^2}$
$0 \cdot \infty$ $\lim_{x \to 0^+} x \ln x$
$\infty - \infty$ $\lim_{x \to 0^+} \left(\frac{1}{x} - \frac{1}{\sin x}\right)$
$0^0$ $\lim_{x \to 0^+} x^x$
$1^\infty$ $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x$
$\infty^0$ $\lim_{x \to \infty} x^{1/x}$

L’Hôpital’s Rule

Statement

If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, and if $\lim_{x \to a} \frac{f'(x)}{g'(x)}$ exists (or is $\pm\infty$), then:

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

Conditions

  1. The limit must be $\frac{0}{0}$ or $\frac{\infty}{\infty}$
  2. $f$ and $g$ must be differentiable near $a$
  3. $g'(x) \neq 0$ near $a$
  4. The limit on the right must exist (or be $\pm\infty$)

Converting Other Forms

$0 \cdot \infty$ Form

Rewrite as $\frac{f}{1/g}$ or $\frac{g}{1/f}$ to get $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

$\infty - \infty$ Form

Find a common denominator or factor to combine into a single fraction.

Exponential Forms ($0^0$, $1^\infty$, $\infty^0$)

Let $y = f(x)^{g(x)}$, then $\ln y = g(x) \ln f(x)$. Find $\lim \ln y$, then $\lim y = e^{\lim \ln y}$.

Examples

Example 1: $\frac{0}{0}$ Form

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

Example 2: $1^\infty$ Form

$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x$$ Let $y = \left(1 + \frac{1}{x}\right)^x$. Then $\ln y = x \ln\left(1 + \frac{1}{x}\right)$. $$\lim_{x \to \infty} \ln y = \lim_{x \to \infty} \frac{\ln(1 + 1/x)}{1/x} = 1$$ Therefore, $\lim y = e^1 = e$.

Common Misconceptions

Common misconception

L’Hopital’s Rule applies to any limit, not only indeterminate forms.

This is the concept-image-conflicts-definition error. The rule requires that the limit be of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ before applying. Applying it to a limit such as $\lim_{x \to 0} \frac{\sin x}{x + 1}$, which equals $\frac{0}{1} = 0$ by direct substitution, gives the wrong answer: $\lim_{x \to 0} \frac{\cos x}{1} = 1 \neq 0$. One must first verify the form is indeterminate; if it is not, direct substitution or algebraic simplification is the correct approach.

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