# Further Maths - L'Hôpital's Rule

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/lhopitals-rule/ · Updated: 2021-11-25 · Tags: school, further-maths, further-pure-1

## See Also
- [Further Maths - Limits](https://ollybritton.com/notes/a-level/further-maths/topics/limits/)

## Flashcards
### 2021-11-25
##### What is L'Hôpital's rule used for??
Finding the limit of two functions divided together.

##### What is $$\lim_{x \to a} \frac{f(x)}{g(x)}$$ equivalent to??
$$
\lim_{x \to a} \frac{f'(x)}{g'(x)}
$$

##### What technique could you use for finding the value of $$\frac{\sin(x)}{x}$$ at $x = 0$??
L'Hôpital's rule.

##### What are the conditions for applying L'Hôpital's rule for $$\lim_{x \to a} \frac{f(x)}{g(x)}$$??
$$
\frac{f(x)}{g(x)} = \frac{0}{0}
$$

or

$$
\frac{f(x)}{g(x)} = \frac{\pm \infty}{\pm \infty}
$$

##### How could you find the limit of the product of two functions $f(x)g(x)$ when their product is undefined??
$$
f(x)g(x) \equiv \frac{g(x)}{1/f(x)} \equiv \frac{f(x)}{1/g(x)}
$$

and use L'Hôpital's rule.

##### How could you evaluate $$\lim_{x \to -\infty} x e^x$$??
$$
\lim_{x \to -\infty} \frac{x}{1/e^x}
$$

##### What do you need to consider when rewriting $f(x)g(x)$ as $\frac{f(x)}{1/g(x)}$ or $\frac{g(x)}{1/f(x)}$ in order to use L'Hôpital's rule??
Which one has a nicer result when you integrate the top and bottom.

##### How could you evaluate $\lim e^{f(x)}$??
$$
e^{\lim f(x)}
$$

##### How can you tackle an indeterminate form like $1^\infty$??
Rewrite as $e^{\infty \times \ln 1}$.

##### How can you tell if a limit doesn't exist??
Approach it from two different directions and see if you get different answers.

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