> For the complete documentation index, see [llms.txt](https://solomons-mathbook.gitbook.io/calculus-basics/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://solomons-mathbook.gitbook.io/calculus-basics/comment-389389887/comment-391295798.md).

# L'Hopital's Rule

`L`Hopital's Rule helps us to find the limit of an `Undefined` limits, like `0/0`, `∞/∞` and such. It's quite simple to apply and very convenient to solve some problems.

[Refer to `L'Hôpital's rule`](https://en.wikipedia.org/wiki/L'Hôpital's_rule)

> [`▶ Back previous note on: Asymptote of Rational Expressions`](https://github.com/solomonxie/solomonxie.github.io/issues/44#issuecomment-374894945) [`▶ Practice at Khan academy: Disguised derivatives`](https://www.khanacademy.org/math/ap-calculus-bc?t=practice)

From my experience, the L'Hopital's Rule is so often been used that we didn't even realize. Actually it's been used almost every time when we are to evaluate the **LIMITS OF RATIONAL EXPRESSIONS**.

![image](https://user-images.githubusercontent.com/14041622/46206943-2e87f300-c358-11e8-8d98-3edc1684ca99.png)

### Example of `0/0`

### Example of `∞/∞`

1. Find the limit: ![image](https://user-images.githubusercontent.com/14041622/40541839-161e29bc-6050-11e8-8847-f627079e4f7e.png) Solve: ![image](https://user-images.githubusercontent.com/14041622/40541894-46865052-6050-11e8-8124-5bfac0cf2141.png)
2. Find the limit: ![image](https://user-images.githubusercontent.com/14041622/40541969-9f1617f2-6050-11e8-9a73-8fa268b14f35.png) Solve: ![image](https://user-images.githubusercontent.com/14041622/40541992-b64715b6-6050-11e8-9627-bb49ea149954.png)

### Example of `1^∞`

## `L'Hopital Rule for Composite functions`

### Example of composite exponential function

![image](https://user-images.githubusercontent.com/14041622/46206901-187a3280-c358-11e8-996b-19f095b40bd8.png) Solve:

* Direct plug in the `x=2` and get `limit = 1^∞`, which is an indeterminate form
* So we're gonna apply the `Log Power Rule` to take down the power:

  ![image](https://user-images.githubusercontent.com/14041622/46207221-2aa8a080-c359-11e8-8080-107a97cda923.png)
* We use **Natural Log** for doing this:

  ![image](https://user-images.githubusercontent.com/14041622/46207262-54fa5e00-c359-11e8-9131-0723617dc762.png)
* And now we can apply the `L'Hopital Rule` for `ln(y)`:

  ![image](https://user-images.githubusercontent.com/14041622/46207374-c2a68a00-c359-11e8-9d56-9258684685ac.png)
* From `ln(y)` we could get `limit of y`:

  ![image](https://user-images.githubusercontent.com/14041622/46207493-292ba800-c35a-11e8-9265-9fa718af66d4.png)