L'Hopital's Rule - Calculus Step by Step

Resolve a 0/0 or ∞/∞ indeterminate form by differentiating the numerator and denominator separately.

Example

Confirm the indeterminate form, differentiate numerator and denominator, then re-evaluate. The step that licenses replacing the ratio by the ratio of derivatives is Cauchy's Mean Value Theorem; this book applies L'Hopital's rule but does not prove it.

highlighted = computed this step

Step 1 — Set up

Set up the limit for L'Hopital's rule.

\lim_{x\to 0 } \frac{\sin x}{x}

Step 2 — Check the form

Check that the form is 0 over 0.

\hlmath{\frac{0}{0}}

Step 3 — Differentiate numerator

Differentiate the numerator to get cosine x.

\frac{d}{dx}\sin x= \hlmath{\cos x}

Step 4 — Differentiate denominator

Differentiate the denominator to get 1.

\frac{d}{dx}x= \hlmath{1}

Step 5 — Re-evaluate

Evaluate cosine 0 over 1 to get 1.

\frac{\cos 0 }{ 1 }= \hl{1}

l-hopital When a limit gives 0/0 or ∞/∞, L'Hopital's rule says: differentiate the numerator and denominator independently, then evaluate the limit of the new fraction.