Then so is the function f k1 x g x k 1 de math

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Unformatted text preview: mits as x → 0. Example. Find lim x→0 sin(x)−x . x2 First notice that this is an indeterminant limit of the type 0 . 0 We know that if f (x) = sin(x), then T1,0 (x) = T 2, 0(x) = x. We will assume that we are working with T2,0 (x). Then Taylor’s Theorem shows that for any x ∈ [−1, 1], there exists a c between 0 and x such that | sin(x) − x |=| − cos(c) 3 1 x |≤ | x |3 3! 6 since | − cos(c) |≤ 1 no matter where c is located. This inequality is equivalent to −1 1 | x |3 ≤ sin(x) − x ≤ | x |3 . 6 6 If x = 0, we can divide all of the terms by x2 to get that for x ∈ [−1, 1] and x = 0 − | x |3 sin(x) − x | x |3 ≤ ≤ 6x2 x2 6x2 or equivalently that −|x| sin(x) − x |x| ≤ ≤ . 6 x2 6 We also know that −|x| |x| = lim =0 x→0 x→0 6 6 The Squeeze Theorem shows that lim sin(x) − x . x→0 x2 lim The technique we outlined in the previous example can be used in much more generality. We will need the following observation: Suppose that f (k+1) (x) is a continuous function on [...
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This document was uploaded on 03/23/2013.

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