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 $f(x),g(x)$ , if $\lim f(x),g(x)\in \{0,\pm \infty\}$ $\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}$ Note that this implies that $\lim\frac{f(x)}{g(x)}=\lim\frac{f^{(n)}(x)}{g^{(n)}(x)}=\lim\frac{f^{(-n)}(x)}{g^{(-n)}(x)}$

Proof

No proof of this theorem is available at this time. You can help AoPSWiki by adding it.

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

Text explanation:

let $z(x) = \frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are both nonzero functions with value $0$ at $x = a$

(for example, $g(x) = cos(\frac{\pi}{2} x)$ , $f(x) = 1-x$ , and $a = 1$ .)

Note that the points surrounding z(a) aren't approaching infinity, as a function like $f(x) = 1/x-1$ might at $f(a)$

The points infinitely close to z(a) will be equal to $\lim{b\to 0} \frac{f(a+b)}{g(a+b)}$

Noting that $\lim{b\to 0} f(a+b)$ and $\lim{b\to 0} g(a+b)$ are equal to $f'(x)$ and $g'(x)$ respectively when $x=a$ . This means that the points approaching $\frac{f(x)}{g(x)}$ at point a where $f(a)$ and $g(a)$ are equal to 0 are equal to $\frac{f'(x)}{g'(x)}$