# L'Hôpital's Rule

**L'Hopital's Rule** is a theorem dealing with limits that is very important to calculus.

## Theorem

The theorem states that for real functions , if Note that this implies that

## Proof

One can prove using linear approximation: The definition of a derivative is which can be rewritten as . Just so all of us know is a function that is both continuous and has a limit of as the in the derivative function approaches . After multiplying the equation above by , we get .

We have already assumed by the hypothesis that the derivative equals zero. Hence, we can rewrite the function as , which would hence prove our lemma for L'Hospital's rule.

Video by 3Blue1Brown: https://www.youtube.com/watch?v=kfF40MiS7zA

Text explanation:

Let , where and are both nonzero functions with value at .

(For example, , , and .)

Note that the points surrounding aren't approaching infinity, as a function like might at .

The points infinitely close to will be equal to .

Note that and are equal to and .

As a recap, this means that the points approaching , where is a number such that and are both equal to , are going to approach .

## Problems

### Introductory

- Evaluate the limit (weblog_entry.php?t=168186 Source)