The theorem states that for real functions , if Note that this implies that
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
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 .
- Evaluate the limit (weblog_entry.php?t=168186 Source)