(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 that <math>f(a)=g(a)=0</math> and that f and g are differentiable on an open interval <i>I</i> containing a. <br> |
| − | + | Suppose also that <math>g'(x)\neq0</math> on <i>I</i> if <math>x\neq a</math>. <br> | |
| − | , | + | Then <br> |
| − | if the | + | <math> |
| + | \lim_{x \to\ a}\frac{f(x)}{g(x)}= \lim_{x \to\ a}\frac{f'(x)}{g'(x)} | ||
| + | </math>, <br> | ||
| + | if the limit on the right exists (or is <math>\infty</math> or -<math>\infty</math> | ||
| + | ). | ||
This is Elizabeth's favorite theorem. | This is Elizabeth's favorite theorem. | ||
Latest revision as of 12:50, 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)\neq0 $ on I if $ x\neq a $.
Then
$ \lim_{x \to\ a}\frac{f(x)}{g(x)}= \lim_{x \to\ a}\frac{f'(x)}{g'(x)} $,
if the limit on the right exists (or is $ \infty $ or -$ \infty $
).
This is Elizabeth's favorite theorem.
