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.

__L’Hopital’s Rule:__If or , where a is finite or ,

Then

**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):

Try Substitution

We get an indeterminate form of

Take the derivative of the numerator:

Take the derivative of the denominator:

Simplify the function:

Substitution:

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,

Leah

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