When Can You Use L'Hôpital's Rule? Limits Explained Step by Step

Short answer: you can use L'Hôpital's rule only when direct substitution gives the indeterminate form 0/0 or ∞/∞. Then lim f(x)/g(x) = lim f′(x)/g′(x), provided the new limit exists. You differentiate the top and bottom separately, not with the quotient rule.

Step 1: Always substitute first

Plug in the value. If you get a real number, that is the answer. If you get a nonzero number over 0, the limit is infinite or does not exist, and L'Hôpital does not apply.

Step 2: Check the form

• 0/0 or ∞/∞: apply the rule.
• 0·∞: rewrite as a fraction first, for example x·ln x = ln x / (1/x).
• ∞ − ∞: combine into one fraction first.
• 1^∞, 0⁰, ∞⁰: take ln, find that limit, then exponentiate.

Worked example: lim_x→0 sin x / x

Substitution gives 0/0. Differentiate top and bottom: cos x / 1 → cos 0 = 1.

Worked example: lim_x→∞ x² / e^x

∞/∞ → 2x / e^x (still ∞/∞) → 2 / e^x → 0. You can apply the rule repeatedly as long as the form stays indeterminate.

Worked example: lim_x→0⁺ x·ln x

This is 0·(−∞). Rewrite as ln x / (1/x), which is −∞/∞. Differentiate: (1/x) / (−1/x²) = −x → 0.

Mistakes that cost marks

• Using the rule when the form is not 0/0 or ∞/∞. This gives wrong answers, not just lost method marks.
• Applying the quotient rule to f/g instead of differentiating f and g separately.
• Not showing the indeterminate form. Many mark schemes require you to state it before applying the rule.

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