c
Math 151 WIR, Spring 2014, 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 2014, Benjamin
Aurispa
√
√
3x + 17 − 5
(d) lim +
x2 − 16
x→−4
(e)
lim
x→−1−
x2 − 7x − 8
x2 + 2x + 1
(f) lim r(t), where r(t) =
t→1
1
1
t−1
,
+
2
t − 4t + 3 t − 1 (t − 2)(t − 1)
2
c
Math 151 WIR, Spring 2014, 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 2014, 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. State if there is any type of continuity
at these locations (i.e. continuity from left/right).
(a) f (x) =
x2 + x − 20
x2 − 16


4x − 10


 x2 − 10
(b) f (x) =

10



2x + 5
if
if
if
if
x≤1
1<x<5
x=5
x>5
4
c
Math 151 WIR, Spring 2014, Benjamin
Aurispa
(c) f (x) =





9
x+3
4x + 3
x4 −2
x−1
if x < 0
if 0 ≤ x < 2
if x ≥ 2
5. For what values of c and d would the following function be continuous everywhere?
f (x) =

2

 x +c
cx + d

 8x − 4d
if x < 3
if 3 ≤ x ≤ 5
if x > 5
6. Use the Intermediate Value Theorem to show that the equation x3 − 3x2 + 1 = 0 has a root on the
interval (2, 3).
5
c
Math 151 WIR, Spring 2014, Benjamin
Aurispa
7. If g(x) = x4 − 2x + 3, show there exists a number c such that g(c) = 9.
8. Calculate the following limits.
(a) lim
x→∞
4x − 5x2 + 2
−6x2 + x − 3
(b)
x→−∞
lim
2x + 3
3 − x2
(c)
lim
2x3 − 9
−3x2 + x
x→−∞
6
c
Math 151 WIR, Spring 2014, Benjamin
Aurispa
9. Find the vertical and horizontal asymptotes (if any) of the following functions.
(a) f (x) =
(b) f (x) =
(c) f (x) =
(x + 5)(x2 − 4)
x2 − 2x − 35
x2
√
x
+1
25x2 + 7
4x − 9
7
c
Math 151 WIR, Spring 2014, Benjamin
Aurispa
10. Calculate the following limits.
√
(a) lim
x2 − 4x − x
x→∞
(b)
lim
x→−∞
p
x2 − 7x + 1 + x
8