L'Hôpital's Rule — Breaking Indeterminate Forms

When to use: Direct substitution gives 0/0 or ∞/∞ (indeterminate forms).

Rule: lim f(x)/g(x) = lim f'(x)/g'(x)

Take derivatives of top and bottom SEPARATELY (not quotient rule).

Example: lim(x→0) sin(x)/x = lim(x→0) cos(x)/1 = 1

Can apply repeatedly: If the result is still 0/0 or ∞/∞, apply L'Hôpital again.

Other indeterminate forms: 0·∞, ∞-∞, 0⁰, ∞⁰, 1^∞ — rewrite as a fraction first, then apply L'Hôpital.

Warning: Only use when you have 0/0 or ∞/∞. If direct substitution gives an actual number, DON'T use L'Hôpital. If you get 5/0, that's not indeterminate — it's ±∞.

Reference:

Wikipedia: L'Hôpital's Rule

https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule