L'Hopital's Rule

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L'Hôpital's Rule

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 function with value $0$ at point $a$

(for example, $g(x) = cos(\frac{\pi}{2} x)$ , $f(x) = 1-x$ , and $a = 0$ .)

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 $\lim{b\to \infty} \frac{f(a+b)}{g(a+b)}$

Noting that $\lim{b\to \infty} f(x+b)$ and $\lim{b\to \infty} g(x+b)$ are equal to $f'(x)$ and $g'(x)$ respectively. 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)}

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L'Hopital's Rule