4.7 Using L'Hopital's Rule for Determining Limits of Indeterminate Forms

Indeterminate forms

When evaluating a limit produces 0/0 or ∞/∞, the result is undetermined — it could be any value, so direct substitution fails. These are called indeterminate forms. Other forms like 0·∞, ∞−∞, 1^∞, 0^0, and ∞^0 can often be algebraically rewritten as 0/0 or ∞/∞ before applying L'Hôpital's Rule.

L'Hôpital's Rule

If lim f(x)/g(x) gives 0/0 or ∞/∞, and f, g are differentiable near the point (g'≠0), then lim f(x)/g(x) = lim f'(x)/g'(x), provided the second limit exists. Take derivatives of numerator and denominator separately (not quotient rule), then re-evaluate. Repeat if still indeterminate.

Checking conditions & applying carefully

Always verify the indeterminate form first — L'Hôpital's Rule cannot be applied to forms like 5/0 or 3/∞, which are evaluated directly. After differentiating, plug in the value again; if still indeterminate, apply the rule again. Misapplying it to non-indeterminate limits gives wrong answers.

Key terms

Indeterminate form

A limit expression like 0/0 or ∞/∞ whose value cannot be determined by direct substitution.

L'Hôpital's Rule

If lim f(x)/g(x) is 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists.

Key requirement before using L'Hôpital's Rule

The limit must first be verified as an indeterminate form (0/0 or ∞/∞).

Common mistake

Using the quotient rule instead of differentiating numerator and denominator separately.

Repeated application

If the new limit is still indeterminate after applying L'Hôpital's Rule once, apply it again.

Try an AP-style question

lim(x→0) (eˣ − 1)/x = ?

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