(New page: 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 \displaystyle\lim_{x\to\a}\fr...) |
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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> | ||
\displaystyle\lim_{x\to\a}\frac{f(x)}{g(x)}=\displaystyle\lim_{x\to\a}\frac{f'(x)}{g'(x)} | \displaystyle\lim_{x\to\a}\frac{f(x)}{g(x)}=\displaystyle\lim_{x\to\a}\frac{f'(x)}{g'(x)} | ||
| − | , | + | </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. | ||
Revision as of 10:13, 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 $ \displaystyle\lim_{x\to\a}\frac{f(x)}{g(x)}=\displaystyle\lim_{x\to\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.
