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**Unformatted text preview: **of the derivatives
2
of cos(x) are everwhere continuous. The Approximation Theorem gives us that there is a
constant M such that
x2
−M | x | ≤ cos(x) − (1 − ) ≤ M | x |3
2
3 for all x ∈ [−1, 1]. Dividing by x2 shows that for all x ∈ [−1, 1] with x = 0
cos(x) − (1 −
−M | x |≤
x2
DE Math 128 332 x2
)
2 ≤M |x|
(B. Forrest)2 CHAPTER 4. Sequences and Series 4.5. Introduction to Taylor Series for all x ∈ [−1, 1]. Simplifying the last expression gives
−M | x |≤ cos(x) − 1 1
+ ≤M |x|
x2
2 for all x ∈ [−1, 1].
The Squeeze Theorem shows that
cos(x) − 1 1
+ =0
x→0
x2
2
lim which is equivalent to
lim x→0 −1
cos(x) − 1
=
.
2
x
2 This limit is consistent with the behaviour of the function h(x) =
information is illustrated in the graph. cos(x)−1
x2 near 0. This The previous limit can actually be calculated quite easily using L’Hˆpital’s Rule. In fact,
o
you should verify the answer using this rule. The next example would require much more
work using L’Hˆpital’s Rule. You will not be responsible for examples that are this como
plicated, but it is provided to show you how powerful Taylor’s Theorem can be for ﬁnding
limits
Example.
Find lim x→0 e x4
2 −cos(x2 )
.
x4 This is also an indeterminant limit of type 0 . We know f...

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