In the previous posts, we have talked about different ways to find the limit of a function. We have gone over factoring, functions with square roots, and rational functions. What happens when none of those options work? That is when L’Hopital’s Rule comes in.
If or , where a is finite or ,
What does L’Hopital’s Rule mean?
If f(x) and g(x) are differentiable and if the limit of as x approaches a is or , then we take the derivative of the numerator and the derivative of the denominator, . The limit of as x approaches a is equal to the limit of as x approaches a.
When can you use L’Hopital’s Rule?
- When the limit is in the indeterminate form of or
- Sometimes when the limit is in the other indeterminate forms
- When the function has an exponent and the limit is an indeterminate form
Here we will work out the first problem step by step (click here):
Take the derivative of the numerator:
Take the derivative of the denominator:
Simplify the function:
Here’s another example (click here):
Last example (click here):
L’Hopital’s rule is an easy way to find the limit, as long as the derivatives aren’t too tedious. It is important to remember to double check your work. I found myself a lot of times making simple errors, like forgetting the chain rule or changing signs. Next week we will talk about the chain rule for limits.
Fun fact: It is believed that Johann (John) Bernoulli discovered L’Hopital’s Rule.
Until next time,