What's 0 ÷ 0? It depends. Some 0/0 limits are 0, some are ∞, some are 1, some are 7. L'Hôpital's rule is how you peer past the indeterminate fog and find the real answer.
When you plug into a limit and get 0/0 or ∞/∞, you can differentiate the top and bottom separately and try again. That's L'Hôpital's rule.
Critical for evaluating limits in physics (small-angle approximations, terminal behaviour), economics (marginal analysis), probability and series convergence.
when both → 0 or both → ∞
lim(x→0) sin(x)/x
Plugging 0 gives 0/0. Differentiate: cos(x)/1. At x=0: cos(0) = 1.
Find lim(x→0) (eˣ − 1) / x.
L'Hôpital only applies to indeterminate forms (0/0, ∞/∞). If the limit is a clean number like 5/2, don't differentiate — you'll get a wrong answer. Always check the form first.
Other indeterminate forms (0·∞, ∞−∞, 1^∞) can be rewritten as 0/0 or ∞/∞ first. Use logs, common denominators, or factoring to get there.
- L'Hôpital's rule replaces an indeterminate 0/0 or ∞/∞ limit with the limit of the derivatives' ratio.
- Always check the form before applying. Apply repeatedly if you still get an indeterminate form.
- Other indeterminate forms can usually be rewritten to 0/0 or ∞/∞ first.