Chapter 1 Worksheet
Fall 2011, Math 1210-007
You may submit your responses to the following review problems to me by 5pm Tuesday, September 6th as an extra credit assignment.
For 1–8, evaluate the following limits.
1.
lim
1 − 2/x
x→2 x2 − 4
5.
1 − cos 2x
x→0
3x
2.
tan x
x→0 sin 2x
6.
sin 5x
x→0 3x
|x|
x
7.
t+2
t→2 (t − 2)2
lim ([[t]] − t)
8.
3.
4.
lim
lim
x→0−
t→2−
lim
lim
lim
lim
x→π/4−
tan 2x
x3 , x < −1
9. Let f (x) =
x, − 1 < x < 1

1 − x, x ≥ 1.


Evaluate the following:
(a)
(b)
f (1)
(c)
lim f (x)
(d)
x→1+
1
lim f (x)
x→1−
lim f (x)
x→−1
(e) What are the values of x at which f is discontinuous?
(f) How should f be defined at x
= −1 to make it continuous there?
10. Use the Intermediate Value Theorem to prove that the equation
2x4 − 15x3 + 28x2 + 9x − 36 = 0
has at least one solution between x
= 1 and x = 2.
x2
.
x2 − 1
11. Find the equations of all vertical and horizontal asymptotes of F (x)
=
12. Find the equations of all vertical and horizontal asymptotes of h(x)
= tan 2x.
13. Find
f (a + h) − f (a)
for the following functions:
(a + h) − a
(a)
f (x) = x2
(b)
f (x) = sin x
2