L'Hopital's Rule: Step-by-Step Guide
L'Hopital's Rule states that for indeterminate forms 0/0 or infinity/infinity, the limit of f(x)/g(x) equals the limit of f'(x)/g'(x), provided the latter exists. It is a powerful shortcut, but it must only be applied after confirming the indeterminate form.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Verify an indeterminate form. Substitute the target value to confirm the limit is 0/0 or infinity/infinity. L'Hopital's rule does NOT apply to forms like 1/0, 0/1, or any defined value.\[\lim_{x \to a} \frac{f(x)}{g(x)} \to \frac{0}{0} \text{ or } \frac{\pm\infty}{\pm\infty}\]
- Differentiate numerator and denominator separately. Compute f'(x) and g'(x). Do NOT use the quotient rule — differentiate top and bottom independently.\[\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\]
- Evaluate the new limit. Try direct substitution on f'(x)/g'(x). If you get a finite value or +/- infinity, that is the answer. If you again get 0/0 or infinity/infinity, apply L'Hopital's rule a second time.\[\lim_{x \to a} \frac{f'(x)}{g'(x)} = L\]
- Convert other indeterminate forms when needed. Forms like 0*infinity, infinity - infinity, 0^0, 1^infinity, and infinity^0 must first be rewritten as 0/0 or infinity/infinity (often using ln and rewriting as a fraction) before L'Hopital applies.\[0 \cdot \infty = \frac{0}{1/\infty} = \frac{0}{0}\]
Worked example
Evaluate lim x->0 (sin x)/x.
Step 1: Substitute x = 0: sin(0)/0 = 0/0, indeterminate, so L'Hopital applies.
Step 2: Differentiate numerator and denominator separately. d/dx[sin x] = cos x and d/dx[x] = 1.
Step 3: The new limit is lim x->0 (cos x)/1.
Step 4: Substitute x = 0: cos(0)/1 = 1/1 = 1.
Step 5: Therefore lim x->0 (sin x)/x = 1.
Verification: At x = 0.01, sin(0.01)/0.01 approximately 0.0099998/0.01 = 0.99998, very close to 1.
Tips and common mistakes
- Always confirm 0/0 or infinity/infinity before applying L'Hopital's rule — using it on a defined limit gives wrong answers.
- Differentiate top and bottom separately — never use the quotient rule.
- You can apply L'Hopital's rule repeatedly if the indeterminate form persists.
- For 0*infinity, rewrite the product as a quotient first: f*g = f/(1/g).
- For infinity - infinity, combine into a single fraction with a common denominator.
- For 0^0, 1^infinity, or infinity^0, take the natural log to convert to a product, then a quotient.
- If after differentiating the new limit gets harder rather than easier, try another method (factoring, series, conjugate).
- L'Hopital does not always succeed: e.g., lim x->infinity x/sqrt(x^2 + 1) cycles forever; use algebra instead.