Limits and Infinity HW
For #1-8, find the vertical asymptotes (if any) of the function.
2)
x2 − 2
h( x ) = 2
x −x−2
x3
x2 −1
4)
g ( x) = tan(2 x)
4
x2
6)
h( x ) =
(8)
s (t ) =
1)
1
f ( x) = 2
x
3)
h( x ) =
5)
f ( x) = 1 −
7)
x3 + 1
m( x ) =
x +1
x
x +x−2
2
t
sin t
For #9-19, find the limit.
9)
lim
x→2
11)
+
lim
x → −3
13)
lim
x→0
15)
x−3
x−2
x 2 + 2x − 3
x2 + x − 6
−
2
sin x
+
lim
x→∞
2x − 1
3x + 2
10)
lim
x → 4+
12)
lim
x→0
14)
lim
x →1
16)
lim
x→∞
x2
x 2 − 16
⎛ 1⎞
⎜1 + ⎟
x⎠
⎝
x2 − x
( x 2 + 1)( x − 1)
x
x −1
2
17)
19)
21)
5x 2
x+3
x → −∞
lim
lim
x→∞
lim
x→∞
x
lim
18)
x → −∞
1
2 x + sin x
20)
ln( x + 1)
ln x
22)
lim
x→∞
lim
x→∞
x2 − x
ln x 2
ln x
ln x
log x
For #23-25, use the graph shown to answer the questions that follow.
23)
(a)
lim
f ( x) =
x → −∞
lim
f ( x) =
x→∞
(c)
What can you conclude about any possible horizontal asymptotes?
(b)
7.0
6.0
5.0
4.0
3.0
2.0
1.0
−9.0
−8.0
−7.0
−6.0
−5.0
−4.0
−3.0
−2.0
−1.0
1.0
−1.0
−2.0
−3.0
−4.0
−5.0
−6.0
−7.0
80
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10
24)
(a)
lim
f ( x) =
x → −∞
(b)
lim
f ( x) =
x→∞
(c)
lim
f ( x) =
−
x → −2
(d)
lim
f ( x) =
x→2
(e)
What can you conclude about any possible horizontal asymptotes?
7.0
6.0
5.0
4.0
3.0
2.0
1.0
−9.0
−8.0
−7.0
−6.0
−5.0
−4.0
−3.0
−2.0
−1.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10
−1.0
−2.0
−3.0
−4.0
−5.0
−6.0
−7.0
80
25)
(a)
lim
f ( x) =
x→0
lim
f ( x) =
x→∞
(b)
2.0
1.0
−4.0
−3.0
−2.0
−1.0
1.0
−1.0
20
2.0
3.0
4.0
5.
NOTE: NO CALCULATOR should be used for #26-30.
26)
27)
The graph of which of the following equations has y = 1 as an asymptote?
(a)
y = cos x
(d)
y=
30)
(e)
y = − ln x
(c)
4
(c)
1
(e)
4x 2 + x − 7
=
x 2 − 5x − 3
x → −∞
0
(b)
7
3
(d)
Nonexistent
ax + b
has a horizontal asymptote y = -2, a vertical asymptote
x+c
x = 4, and an x-intercept of 1.5, then a – b + c =
If the graph of y =
(a)
29)
y=e
x3
y= 2
x +1
lim
(a)
28)
x2
x2 − 5
(b)
x
-3
(b)
1
(c)
5
(d)
-9
(e)
-1
(c)
III only
4x 2 − 3
is
The function f ( x) =
2x 2 + 1
I.
II.
III.
IV.
unbounded
bounded below by y = -3
bounded above by y = 2
bounded below by y = 2
(a)
I
(b)
IV only
(d)
II and III
(e)
II only
If f ( x) = e x + 2 , which of the following lines is an asymptote to the graph of f?
(a)
y = -2
(b)
x=0
(d)
x=2
(e)
y=0
(c)
y=2