L'Hopital's Rule feels like a magic shortcut once you learn it — which is exactly why it gets misused. It only works on a specific type of limit, and applying it when it doesn't apply is one of the most common point-losing mistakes on a calculus exam.
When it actually applies
L'Hopital's Rule only applies to limits that are in an indeterminate form — specifically 0/0 or ∞/∞. If you plug in the limit value and get a real number (even 0 divided by a nonzero number), you don't need L'Hopital's Rule — you already have your answer.
The step-by-step method
- Plug in the limit value first to confirm you actually get 0/0 or ∞/∞. This step is not optional.
- Take the derivative of the numerator and denominator separately — not the derivative of the whole fraction using the quotient rule.
- Evaluate the new limit. If it resolves to a number, you're done.
- If it's still indeterminate, apply the rule again on the new fraction, and repeat until it resolves.
Worked example
Evaluate the limit as x → 0 of (sin x) / x
Plugging in x = 0 gives 0/0 — an indeterminate form, so L'Hopital's Rule applies.
Differentiate top and bottom separately: derivative of sin x is cos x; derivative of x is 1.
New limit: cos(0) / 1 = 1
So the original limit equals 1.
Common mistakes
Applying it to a limit that isn't actually indeterminate. If plugging in gives something like 5/0 (not 0/0), the limit is infinite or doesn't exist — L'Hopital's Rule doesn't apply and using it will give a wrong answer.
Using the quotient rule instead of differentiating top and bottom separately. L'Hopital's Rule is NOT the quotient rule — you take the derivative of the numerator and denominator independently, not the derivative of the fraction as a whole.
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