Difference between revisions of "L'Hôpital's Rule"
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| − | 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 | + | 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 <math>x = a</math> |
| − | (for example, <math>g(x) = cos(\frac{\pi}{2} x)</math>, <math>f(x) = 1-x</math>, and <math>a = | + | (for example, <math>g(x) = cos(\frac{\pi}{2} x)</math>, <math>f(x) = 1-x</math>, and <math>a = 1</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> | 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> | ||
Revision as of 20:29, 11 March 2022
L'Hopital's Rule is a theorem dealing with limits that is very important to calculus.
Theorem
The theorem states that for real functions 
, if 
![\[\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}\]](http://latex.artofproblemsolving.com/6/7/6/67664051ac365e271260749b981666330b2b71af.png)
Note that this implies that ![\[\lim\frac{f(x)}{g(x)}=\lim\frac{f^{(n)}(x)}{g^{(n)}(x)}=\lim\frac{f^{(-n)}(x)}{g^{(-n)}(x)}\]](http://latex.artofproblemsolving.com/f/6/6/f66ad9bb5b49e51afc1bf4dac34285dae981fd6f.png)
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 
where 
and 
are both nonzero function with value 
at 
(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)}
Problems
Introductory
- Evaluate the limit
(weblog_entry.php?t=168186 Source)
(
