L'Hôpital's Rule & Indeterminate Forms Cheat Sheet

A printable reference covering indeterminate forms, L'Hôpital's Rule, limits, derivatives, and algebraic rewrites for grades 11-12.

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This cheat sheet covers L'Hôpital's Rule and the main indeterminate forms that appear in calculus limits. Students need it because many difficult limits cannot be solved by direct substitution alone. It gives a clear process for recognizing when the rule applies and when an expression must be rewritten first. It also helps students avoid using L'Hôpital's Rule when simpler algebra or standard limits are better.

Key Facts

L'Hôpital's Rule applies to limits of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ when $\lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)}$ exists.
Before using L'Hôpital's Rule, direct substitution should give an indeterminate form such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$ .
For products of the form $0 \cdot \infty$ , rewrite as a quotient such as $f(x)g(x)=\frac{f(x)}{1/g(x)}$ or $\frac{g(x)}{1/f(x)}$ .
For differences of the form $\infty - \infty$ , combine terms using a common denominator, conjugate, or algebraic simplification before applying L'Hôpital's Rule.
For powers of the forms $0^0$ , $1^{\infty}$ , and $\infty^0$ , use logarithms by setting $y = [f(x)]^{g(x)}$ and studying $\ln y = g(x)\ln(f(x))$ .
L'Hôpital's Rule may be applied more than once if each new quotient still has the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ .
L'Hôpital's Rule does not mean differentiating the entire quotient, so $\frac{f(x)}{g(x)}$ becomes $\frac{f'(x)}{g'(x)}$ , not $\left(\frac{f(x)}{g(x)}\right)'$ .
If a simpler method works, such as factoring, rationalizing, or using $\lim_{x \to 0}\frac{\sin x}{x}=1$ , that method is often faster and safer.

Vocabulary

Indeterminate form: An expression such as $\frac{0}{0}$ or $\infty - \infty$ whose limiting value cannot be determined from substitution alone.
L'Hôpital's Rule: A limit rule that allows certain quotients to be evaluated by replacing $\frac{f(x)}{g(x)}$ with $\frac{f'(x)}{g'(x)}$ .
Direct substitution: The method of evaluating a limit by replacing $x$ with the value it approaches, such as $x=a$ .
One-sided limit: A limit in which $x$ approaches a value from only the left or the right, written $\lim_{x \to a^-}f(x)$ or $\lim_{x \to a^+}f(x)$ .
Logarithmic transformation: A method for power limits that uses $\ln y$ to turn exponents into products, especially for $0^0$ , $1^{\infty}$ , and $\infty^0$ .
Rationalizing: An algebraic method that multiplies by a conjugate to simplify expressions with radicals, such as $\sqrt{x+1}-1$ .

Common Mistakes to Avoid

Using L'Hôpital's Rule without checking the form is wrong because the rule only applies directly to $\frac{0}{0}$ or $\frac{\infty}{\infty}$ quotients.
Differentiating the quotient as $\left(\frac{f(x)}{g(x)}\right)'$ is wrong because L'Hôpital's Rule requires the quotient $\frac{f'(x)}{g'(x)}$ .
Applying the rule to $0 \cdot \infty$ without rewriting is wrong because products must first be converted into a quotient form.
Stopping after one use when the result is still $\frac{0}{0}$ is wrong because L'Hôpital's Rule may need to be applied repeatedly.
Ignoring domain and one-sided behavior is wrong because expressions involving $\ln x$ , radicals, or vertical asymptotes may only be valid from one side.

Practice Questions

1 Evaluate $\lim_{x \to 0}\frac{\sin(3x)}{x}$ .
2 Evaluate $\lim_{x \to \infty}\frac{2x^2+5x}{e^x}$ .
3 Evaluate $\lim_{x \to 0^+}x\ln x$ by first rewriting it as a quotient.
4 Explain why L'Hôpital's Rule cannot be applied directly to $\lim_{x \to 0}(\frac{1}{x}-\frac{1}{\sin x})$ before rewriting the expression.

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