c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
Math 151 Week in Review 3
Sections 2.3, 2.5, 2.6
p
3xf (x) g(x)
1. Given that lim f (x) = 2, lim g(x) = 4, and lim h(x) = 5, calculate lim
x→3
x→3
x→3
x→3 (x − 1)h(x)
2. Calculate the following limits or state why the limit does not exist.
√
x+2
(a) lim
x→4 x − 1
(3 + h)2 + 2(3 + h) − 15
h→0
h
(b) lim
(c) lim
x→2
x
2x+1
− 25
x−2
1
c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
√
(d) lim
x→−4
3x + 17 −
x2 − 16
√
5
x2 − 7x − 8
x→−1 x2 + 2x + 1
(e) lim
(f) lim r(t), where r(t) =
t→1
t−1
1
1
,
+
2
t − 4t + 3 t − 1 (t − 2)(t − 1)
2
c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
x2 − 9
x→−3 |x + 3|
(g) lim
(h) lim f (x) where f (x) =
x→−2
(i) lim x6 cos
x→0


 3x + 1
3

 x2 − 9
if x < −2
if x = −2
if x > −2
1
x
3
c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
3. Given that 3x − 7 ≤ f (x) ≤ x3 − 3x2 + 3x − 3 for all x where 1 ≤ x ≤ 3, calculate lim f (x).
x→2
4. Determine where the following functions are not continuous and explain why mathematically. At the
values for which the function is not continuous state whether the function is continuous from the right,
left, or neither.
(a) f (x) =
(b) f (x) =
x2 + x − 20
x2 − 16

9

 x+3
4x + 3

 x4 − 2
if x < 0
if 0 ≤ x < 2
if x ≥ 2
4
c
Math 151 WIR, Spring 2010, Benjamin
Aurispa


4x − 10


 x2 − 10
(c) f (x) =

10


 4x+10
if
if
if
if
x−3
x≤1
1<x<5
x=5
x>5
5. Which of the following functions has a removable discontinuity? If removable at a, find a function g
that agrees with f for all x 6= a and is continuous at a.
(a) f (x) = 3x2 + πx −
1
e
(b) f (x) =
(x − 2)(x2 − 6x − 27)
x+3
(c) f (x) =
(x − 2)(x + 3)
x−6
5
c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
6. For what values of c and d would the following function be continuous?
f (x) =

2

 x +c
cx + d

 8x − 4d
if x < 3
if 3 ≤ x ≤ 5
if x > 5
7. Use the Intermediate Value Theorem to show that the equation x3 − 3x2 + 1 = 0 has a root on the
interval (2, 3).
8. If g(x) = x4 − 2x + 3, show there exists a number c such that g(c) = 9.
6
c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
9. Calculate the following limits.
(a) lim
x→∞
(b)
lim
x→−∞
(c) lim
x→∞
4x − 5x2 + 2
−6x2 + x − 3
2x + 3
3 − x2
√
25x2 − 7x
4x − 1
7
c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
(d) lim
x→∞
(e)
lim
x→−∞
√
x2 − 4x − x
p
x2 − 7x + 1 + x
8
c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
10. Find all vertical and horizontal asymptotes of the following functions.
(a)
1 − 2x
9x + 1
(b)
(x + 5)(x2 − 4)
x2 − 2x − 35
(c) √
x+5
5x2 + 4
9