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**Unformatted text preview: **r two inequalities together to get a rather messy inequality:
−( x4
x4
M1
M1
x4
+ M2 )x8 ≤ (e 2 − (1 + ) − (cos(x2 ) − (1 − )) ≤ (
+ M2 )x8
4
2
2
4 However, if we let M = M1
4 + M2 and simplify, this inequality becomes
x4 −M x8 ≤ e 2 − cos(x2 ) − x4 ≤ M x8
for all x ∈ [−1, 1]. Dividing by x4 gives us that
x4 e 2 − cos(x2 )
− 1 ≤ M x4
−M x ≤
x4
4 The ﬁnal step is to use the Squeeze Theorem to show that
x4 e 2 − cos(x2 )
−1=0
lim
x→0
x4
or equivalently that
x4 e 2 − cos(x2 )
= 1.
lim
x→0
x4
This limit is conﬁrmed by the graph of the function h(x) =
DE Math 128 334 e x4
2 −cos(x2 )
.
x4 (B. Forrest)2 CHAPTER 4. Sequences and Series 4.5. Introduction to Taylor Series This is a rather complicated looking argument. However, with a little practice using Taylor
polynomials and the mastery of a few tricks, limits like this can actually be done by
inspection! 4.5.3 Convergence of Taylor Series In this section we will return to a question that we asked earlier:
Question: Given a functionf (x) that is inﬁnitely diﬀerentiable at x = a, is f (x) equal t...

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