Difference between revisions of "L'Hôpital's Rule"
m (More cosmetics I didn't notice earlier.) |
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Note that this implies that <cmath>\lim\frac{f(x)}{g(x)}=\lim\frac{f^{(n)}(x)}{g^{(n)}(x)}=\lim\frac{f^{(-n)}(x)}{g^{(-n)}(x)}</cmath> | Note that this implies that <cmath>\lim\frac{f(x)}{g(x)}=\lim\frac{f^{(n)}(x)}{g^{(n)}(x)}=\lim\frac{f^{(-n)}(x)}{g^{(-n)}(x)}</cmath> | ||
==Proof== | ==Proof== | ||
− | :'' | + | One can prove using linear approximation: |
+ | The definition of a derivative is <math>f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}</math> which can be rewritten as <math>f'(x) = \frac{f(x+h)-f(x)}{h} + \eta(h)</math>. | ||
+ | Just so all of us know <math>\eta(h)</math> is a function that is both continuous and has a limit of <math>0</math> as the <math>h</math> in the derivative function approaches <math>0</math>. | ||
+ | After multiplying the equation above by <math>h</math>, we get <math>f(x+h) = f(x) +f'(x)h+h\cdot \eta(h)</math>. | ||
+ | |||
+ | We have already assumed by the hypothesis that the derivative equals zero. Hence, we can rewrite the function as <math>\frac{f(x_0+h)}{g(x_0+h)}=\frac{f'(x_0)h + h\cdot \eta(h)}{g'(x_0)h+h\cdot \epsilon(h)}</math>, which would hence prove our lemma for L'Hospital's rule. | ||
+ | |||
Video by 3Blue1Brown: https://www.youtube.com/watch?v=kfF40MiS7zA | Video by 3Blue1Brown: https://www.youtube.com/watch?v=kfF40MiS7zA |
Latest revision as of 14:04, 24 March 2022
L'Hopital's Rule is a theorem dealing with limits that is very important to calculus.
Contents
[hide]Theorem
The theorem states that for real functions , if Note that this implies that
Proof
One can prove using linear approximation: The definition of a derivative is which can be rewritten as . Just so all of us know is a function that is both continuous and has a limit of as the in the derivative function approaches . After multiplying the equation above by , we get .
We have already assumed by the hypothesis that the derivative equals zero. Hence, we can rewrite the function as , which would hence prove our lemma for L'Hospital's rule.
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 aren't approaching infinity, as a function like might at .
The points infinitely close to will be equal to .
Note that and are equal to and .
As a recap, this means that the points approaching , where is a number such that and are both equal to , are going to approach .
Problems
Introductory
- Evaluate the limit (weblog_entry.php?t=168186 Source)