Section 4.3: L'Hopital's Rule

0/0 Indeterminate Forms: Version 1

For c a real number, if

limx → cf(x) = 0andlimx → cg(x) = 0

then:

limx → cf(x)/g(x) = limx → cf'(x)/g'(x)

provided that the second limit exists or is ∞ or -∞.

0/0 Indeterminate Forms: Version 2 (one-sided limits and limits at infinity)

The first version of L'Hopital's rule remains valid if the symbol x → c is replaced everywhere it occurs with one of the following symbols:

1. x → c⁺
2. x → c^-
3. x → ∞
4. x → -∞

∞/∞ Indeterminate Forms: Version 3

Version 1 and 2 of L'Hopital's rule for the indeterminate form 0/0 are also valid if the limit of f and the limit of g are both infinite; that is, both +∞ and -∞ are permissible for either limit.