Difference between revisions of "L'Hôpital's Rule"
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The points infinitely close to z(a) will be equal to <math>\lim{b\to 0} \frac{f(a+b)}{g(a+b)}</math> | The points infinitely close to z(a) will be equal to <math>\lim{b\to 0} \frac{f(a+b)}{g(a+b)}</math> | ||
− | Noting that <math>\lim{b\to 0} f(a+b)</math> and <math>\lim{b\to 0} g(a+b)</math> are equal to <math>f'( | + | Noting that <math>\lim{b\to 0} f(a+b)</math> and <math>\lim{b\to 0} g(a+b)</math> are equal to <math>f'(a)</math> and <math>g'(a)</math>. |
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 <math>\frac{f'(x)}{g'(x)}</math> | 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 <math>\frac{f'(x)}{g'(x)}</math> | ||
Revision as of 18:39, 13 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 Note that this implies that
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 functions 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 . This means that the points approaching at point a where and are equal to 0 are equal to
Problems
Introductory
- Evaluate the limit (weblog_entry.php?t=168186 Source)