L Hopital Rule and Indeterminate Forms Cheat Sheet

Key Facts

• L’Hôpital’s Rule applies to limits of the form $\lim_{x \to a} \frac{f(x)}{g(x)}$ when direct substitution gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
• If the conditions are met, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, provided the derivative limit exists or is infinite.
• For limits at infinity, L’Hôpital’s Rule may be used on $\lim_{x \to \infty} \frac{f(x)}{g(x)}$ or $\lim_{x \to -\infty} \frac{f(x)}{g(x)}$ if the form is $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
• The expression $0 \cdot \infty$ is not a quotient form, so rewrite it as $\frac{f(x)}{1/g(x)}$ or $\frac{g(x)}{1/f(x)}$ before applying L’Hôpital’s Rule.
• The expression $\infty - \infty$ should often be combined into one fraction using algebra, conjugates, or common denominators before using L’Hôpital’s Rule.
• For power forms, use logarithms by setting $y = f(x)^{g(x)}$, then analyze $\ln y = g(x)\ln(f(x))$.
• L’Hôpital’s Rule differentiates the numerator and denominator separately, so $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)$ is not used.
• If repeated use still gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$, L’Hôpital’s Rule may be applied again as long as the conditions continue to hold.

Vocabulary

Indeterminate form

An expression such as $\frac{0}{0}$ or \infty - \infty} whose limit cannot be determined from substitution alone.

L’Hôpital’s Rule

A theorem that allows certain quotient limits to be evaluated using $\lim_{x \to a} \frac{f'(x)}{g'(x)}$ instead of $\lim_{x \to a} \frac{f(x)}{g(x)}$.

Quotient form

A limit written as a ratio $\frac{f(x)}{g(x)}$, which is required before directly applying L’Hôpital’s Rule.

Limit at infinity

A limit that studies the behavior of a function as $x \to \infty$ or $x \to -\infty$.

Derivative quotient

The expression $\frac{f'(x)}{g'(x)}$ formed by differentiating the numerator and denominator separately.

Logarithmic transformation

A method for handling power indeterminate forms by rewriting $y = f(x)^{g(x)}$ as $\ln y = g(x)\ln(f(x))$.