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Suppose that f(a)=g(a)=0 and that f and g are differentiable on an open interval <i>I</i> containing a. Suppose also that g'(x)/=0 on <i>I</i> if x/=a. Then | Suppose that f(a)=g(a)=0 and that f and g are differentiable on an open interval <i>I</i> containing a. Suppose also that g'(x)/=0 on <i>I</i> if x/=a. Then | ||
<math> | <math> | ||
| − | + | \lim_{x -> a}\frac{f(x)}{g(x)}= \lim_{x-> a}\frac{f'(x)}{g'(x)} | |
</math>, | </math>, | ||
if the limis on the right exists (or is positive or negative infinity). | if the limis on the right exists (or is positive or negative infinity). | ||
This is Elizabeth's favorite theorem. | This is Elizabeth's favorite theorem. | ||
| + | |||
| + | |||
| + | <math> | ||
| + | \displaystyle\lim_{x\to\a}\frac{f(x)}{g(x)}=\displaystyle\lim_{x\to\a}\frac{f'(x)}{g'(x)} | ||
| + | </math>, | ||
Revision as of 11:40, 4 September 2008
Suppose that f(a)=g(a)=0 and that f and g are differentiable on an open interval I containing a. Suppose also that g'(x)/=0 on I if x/=a. Then $ \lim_{x -> a}\frac{f(x)}{g(x)}= \lim_{x-> a}\frac{f'(x)}{g'(x)} $, if the limis on the right exists (or is positive or negative infinity).
This is Elizabeth's favorite theorem.
$ \displaystyle\lim_{x\to\a}\frac{f(x)}{g(x)}=\displaystyle\lim_{x\to\a}\frac{f'(x)}{g'(x)} $,
