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Unformatted text preview: INDETERMINATE FORMS AND L’HOSPITAL’S RULE We studied the differentiation and its applications. As another application, we will see how we can use the differentiation to calculate limits. Let’s consider the following example. Example 1. Find lim x → e x- 1 2 x . We actually have a quotient rule for limits, namely lim x → f ( x ) g ( x ) = lim x → f ( x ) lim x → g ( x ) . However, if we apply this rule to the example, we get “ ”. We call this type of limits “indeterminate form” of “type ”. If we draw the graphs of y = e x- 1 and y = 2 x , we can see that, near the origin, those graphs can be approximated very well by their tangent lines. Thus, we can guess that lim x → e x- 1 2 x = lim x → x 2 x = 1 2 . We can generalize this result by saying that the limit of quotient is equal to the quotient of slopes. We have the following. Theorem 1. (l’Hospital’s rule - simple version) Suppose f ( x ) and g ( x ) are differ- entiable and lim x → a f ( x ) = 0 , lim x → a g ( x ) = 0 , g ( a ) 6 = 0 . Then, lim x → a f ( x ) g ( x ) = f ( a ) g ( a ) , if the limit on the right hand side exists (or is ∞ or-∞...
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- Fall '09
- Limits, lim, Limit of a function, Indeterminate form, 1 sec