L'Hopital's Rule
Read this section to learn how to use and apply L'Hopital's Rule. Work through practice problems 1-3.
A Linear Example
Two linear functions are given in Fig. 1 , and we need to find . Unfortunately, and so we cannot apply the Main Limit Theorem. However, we know and are linear, we can calculate their slopes from Fig. 1 , and we know that they both go through the point so we can find their equations: and .
Fig. 1
In fact, this pattern works for any two linear functions:
The really powerful result, discovered by John Bernoulli and named for the Marquis de 1'Hô pital who published it in his calculus book, is that the same pattern is true for differentiable functions even if they are not linear.
Idea for a proof: Even though and may not be linear functions, they are differentiable so at the point they are "almost linear" in the sense that they are well approximated by their tangent lines at that point (Fig. 2):
Fig. 2
(Unfortunately, we have ignored a couple subtle difficulties such as or possibly being when is close to . A proof of Hô pital's Rule is difficult and is not included.)
Example 1: Use 1'Hô pital's Rule to determine and .
Solution: (a) We could evaluate this limit without 1'Hô pital's Rule but let's use it. We can match the pattern of 1 'Hô pital's Rule by letting and . Then , and and are differentiable with and so
(b) Let and . Then and are differentiable for near , and and . Then