2.2.11-eg-1 - zero. However, factoring leads to lim h →...

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Example Evaluate the limit lim h 0 f ( x + h ) - f ( x ) h for the function f ( x ) = x 3 and the point x = 2 . Solution: Using x = 2, f ( x + h ) = f (2 + h ) = (2 + h ) 3 = 2 3 + 3 · 2 2 · h + 3 · 2 · h 2 + h 3 , i.e., f (2 + h ) = 8 + 12 h + 6 h 2 + h 3 . It then follows that lim h 0 f ( x + h ) - f ( x ) h = lim h 0 f (2 + h ) - f (2) h = lim h 0 (8 + 12 h + 6 h 2 + h 3 ) - 8 h = lim h 0 12 h + 6 h 2 + h 3 h , where we cannot simply evaluate the last expression at h = 0 because we will be dividing by
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Unformatted text preview: zero. However, factoring leads to lim h → 12 h + 6 h 2 + h 3 h = lim h → (12 + 6 h + h 2 ) h h = lim h → (12 + 6 h + h 2 ) · h h = lim h → (12 + 6 h + h 2 ) = 12 , where we have used the fact that h h = 1 , for all h 6 = 0 ....
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

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