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```									ESP – Math 179                      Worksheet 3                            Spring 2010

1. Assume that −x4 ≤ τ (x) ≤ x2 . What is limx→0 τ (x)? Is there enough information given
to evaluate limx→−1 τ (x)?

2. Use the Squeeze Theorem from Section 2.6 to evaluate the limit.
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(a) limx→0 x cos x
√      π
(b) limx→0+ xecos( x )

3. Use the Squeeze Theorem to show that if limx→c |g(x)| = 0, then limx→c g(x) = 0.

4. Use Theorem 2 in Section 2.6 to evaluate the limit.
tan x
(a) limx→0    x
√
(b)   limx→0 x+2 sin x
x
ex sin x
(c) limx→0       x

5. Evaluate the limit.
6 sin x
(a) limx→0      6x
x
(b) limx→ π
4   sin 11x

1−cos x
(c) limx→0     sin x

6. Use the Intermediate Value Theorem from Section 2.7 to show that

ex + ln x = 0

has a solution in the interval (0, 1).

7. Let µ(x) = 2x − x3 .

(a) Show that µ(x) has a zero in the interval [1, 1.5].
(b) Show that µ(x) has a zero in the interval [1.25, 1.5].
(c) Determine whether [1.25, 1.375] or [1.375, 1.5] contains a zero.

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8. In Chapter 2, you learned several diﬀerent ways to tell if a limit exists and (if it does)
to evaluate limits. Use the methods of Chapter 2 to evaluate the following limits:
x3 −x
(a) limx→1    x−1

e3x −ex
(b) limx→0     ex −1

x3 −ax2 +ax−1
(c) limx→1         x−1

[x]
(d) limx→0−     x
sin 5x
(e) limx→0  sin 2x
√
(f)   limx→3 x+1−2
x−3

9. Deﬁne the derivative of f (x) at x = a (denoted by f (x)) by the following limit (if it
exists):

f (a + h) − f (a)
f (a) = lim                   .
h→0         h
For the following functions, compute f (x) and f (0).

(a) f (x) = x2 − x + 4

(b) f (x) = x3 + 2x − 1
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(c) f (x) =   x

(d) f (x) = sin x (Hint: Look up the sum and diﬀerence formulas for sin x for help)

(e) f (x) = ex

10. We interpret the derivative of f (x) at x = a as being the instantaneous slope of the
graph at the point x = a. Let f (x) = |x|. Can f (0) exist? Why or why not? (Hint: Graph
the function, and think about the instantaneous slope of the graph to the left of 0 and to
the right of 0.)

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