4.7 L'Hospital's Rule

When it applies

For an indeterminate form 0/0 or ∞/∞: lim f/g = lim f'/g' (when the new limit exists). Always check the form first — L'Hospital's Rule only applies to those indeterminate forms.

How to use it

Differentiate the numerator and denominator separately (this is NOT the quotient rule), then take the limit. Example: lim(x→0) sin x / x = lim cos x / 1 = 1.

Key terms

L'Hospital's Rule

For 0/0 or ∞/∞, lim f/g = lim f'/g'.

Check first

Only for indeterminate forms 0/0 or ∞/∞.

How

Differentiate numerator and denominator separately.

Not the quotient rule

You do NOT use the quotient rule here.

Example

lim(x→0) sin x / x = 1.

Try an AP-style question

lim(x→0) (eˣ − 1)/x = ?

1. e
2. Does not exist
3. 0
4. 1