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# lec34 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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Lecture 34 18.01 Fall 2006 Lecture 34: Indeterminate Forms - L’Hôpital’s Rule L’Hôpital’s Rule (Two correct spellings: “L’Hôpital” and “L’Hospital”) Sometimes, we run into indeterminate forms. These are things like 0 0 and For instance, how do you deal with the following? lim x 3 1 = 0 ?? x 1 x 2 1 0 Example 0. One way of dealing with this is to use algebra to simplify things: lim x 3 1 = lim ( x 1)( x 2 + x + 1) = lim x 2 + x + 1 = 3 x 1 x 2 1 x 1 ( x 1)( x + 1) x 1 x + 1 2 In general, when f ( a ) = g ( a ) = 0 , f ( x ) f ( x ) x lim a f ( x ) f ( a ) f ( a ) lim = lim x a = x a = x a g ( x ) x a g ( x ) lim g ( x ) g ( a ) g ( a ) x a x a x a This is the easy version of L’Hôpital’s rule: f ( x ) f ( a ) lim = x a g ( x ) g ( a ) Note: this only works when g ( a ) = 0 ! In example 0, f ( x ) = x 3 = 1; g ( x ) = x 2 1 f ( x ) = 3 x 2 ; g ( x ) = 2 x = f (1) = 3; g (1) = 2 The limit is f (1) /g (1) = 3 / 2 . Now, let’s go on to the full L’Hôpital rule.
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