Sivakumar
MATH 171
Exercise Set 3
From the text:
Page
Page
Page
Page
99: 15–19, 22, 26, 32, 34, 37, 38
100: 56, 64
101: 76, 78, 79
133: 4, 5, 6, 12, 13, 14–18, 24, 26, 28, 30
x2 − 1
.
|x − 1|
(i) Sketch the graph of F .
(ii) Find lim− F (x).
1. Let F (x) =
x→1
(iii) Find lim+ F (x).
x→1
(iv) Does lim F (x) exist? Explain.
x→1
2. Find numbers a and b such that
x2 + ax − b
= 2.
x→a
x−a
lim
The next pair of examples is taken from the book Basic Analysis: Japanese Grade 11, translated
and published by the American Mathematical Society, as part of the University of Chicago School
Mathematics Project.
3. Find the values of the constants a and b such that
x2 − ax + 8
1
= .
2
x→2 x − (b + 2)x + 2b
5
lim
4. Find the cubic function f (x) (to wit, a function of the form f (x) = ax3 + bx2 + cx + d, where
a, b, c, and d are constants) which satisfies the following conditions:
f (x)
=2
x→0 x
lim
and
f (x)
= 1.
x→1 x − 1
lim
5. (i) Give a proof of the Sandwich Principle/Squeeze Theorem stated and discussed in lecture.
(ii) Prove the following version of the Squeeze Theorem: Suppose that f (x) ≤ g(x) ≤ h(x) for
every x > A (where A is some fixed real number). If lim f (x) = lim h(x) = L (where L is a
x→∞
real number), then lim g(x) = L as well.
x→∞
sin2 x
.
x→∞ x2
(iii) Use the theorem from part (ii) to evaluate lim
1
x→∞