Difference between revisions of "L'Hopital's Rule"
(L'Hopital's Rule moved to L'Hôpital's Rule: It should be l'Hôpital or l'Hospital) |
(Proof by Intuition) |
||
| Line 1: | Line 1: | ||
#REDIRECT [[L'Hôpital's Rule]] | #REDIRECT [[L'Hôpital's Rule]] | ||
| + | |||
| + | Video by 3Blue1Brown: https://www.youtube.com/watch?v=kfF40MiS7zA | ||
| + | |||
| + | Text explanation: | ||
| + | |||
| + | let <math>z(x) = \frac{f(x)}{g(x)}</math> where <math>f(x)</math> and <math>g(x)</math> are both nonzero function with value <math>0</math> at point <math>a</math> | ||
| + | |||
| + | (for example, <math>g(x) = cos(\frac{\pi}{2} x)</math>, <math>f(x) = 1-x</math>, and <math>a = 0</math>.) | ||
| + | |||
| + | Note that the points surrounding z(a) aren't approaching infinity, as a function like <math>f(x) = 1/x-1</math> might at <math>f(a)</math> | ||
| + | |||
| + | The points infinitely close to z(a) will be equal to <math>\lim{b\to \infty} \frac{f(a+b)}{g(a+b)}</math> | ||
| + | |||
| + | Noting that <math>\lim{b\to \infty} f(x+b)</math> and <math>\lim{b\to \infty} g(x+b)</math> are equal to <math>f'(x)</math> and <math>g'(x)</math> respectively. | ||
| + | This means that the points approaching <math>\frac{f(x)}{g(x)}</math> at point a where <math>f(a)</math> and <math>g(a)</math> are equal to 0 are equal to $\frac{f'(x)}{g'(x)} | ||
Revision as of 21:21, 11 March 2022
Redirect to:
Video by 3Blue1Brown: https://www.youtube.com/watch?v=kfF40MiS7zA
Text explanation:
let 
where 
and 
are both nonzero function with value 
at point 
(for example, 
, 
, and 
.)
Note that the points surrounding z(a) aren't approaching infinity, as a function like 
might at 
The points infinitely close to z(a) will be equal to 
Noting that 
and 
are equal to 
and 
respectively.
This means that the points approaching 
at point a where and 
are equal to 0 are equal to $\frac{f'(x)}{g'(x)}
