L’HÔPITAL’S RULE
L’Hôpital’s Rule is a way to find the limit of certain kinds of expressions that are indeterminate forms. If
the limit of an expression results in or , the limit is called “indeterminate” and you can use
L’Hôpital’s Rule to evaluate these expressions.
If f(c) = g(c) = 0, and if f′(c) and g′(c) exist, and if g′(c) ≠ 0, then .Similarly,
If f(c) = g(c) = ∞, and if f′(c) and g′(c) exist, and if g′(c) ≠ 0, then .In other words, if the limit of the function gives us an undefined expression, like or , L’Hôpital’s Rule
says we can take the derivative of the top and the derivative of the bottom and see if we get a determinate
expression. If not, we can repeat the process.
Example 1: Find .
First, notice that plugging in 0 results in , which is indeterminate. Take the derivative of the top and of
the bottom.
The limit equals 1.
Example 2: Find .