10.2: Infinite Limits
Infinite Limits
• When the limit of f(x) does not exist and
f(x) goes to positive infinity or negative
infinity, then we can call that an infinite
limit.
Discuss the behavior of
f ( x) 
1
x 1
as x 1
.9
.99
.999
.9999
1
1.0001
1.001 1.01
1.1
-10
-100
-1000
-10000 ?
10,000
1000
10
• As x goes closer to 1 from
the left, f(x) is smaller and
smaller
• As x goes closer to 1 from
the right, f(x) is bigger and
bigger
100
lim
1
 
x 1
lim
1

x 1
lim
1
does _ not _ exist
x 1
x 1
x 1
x 1
Describe the behavior of
x2  x  2
f ( x) 
x2 1
at 1 and -1
.9
.99
.999
1
1.001
1.01
1.1
1.5263
1.5025126
1.5002501
?
1.4997501
1.4975124
1.4761905
-1.1
-1.01
-1.001
-1
-0.999
-0.99
-0.9
-9
-99
-999
?
1001
101
11
x2  x  2
lim 
 
2
because x  1
x 1
x2  x  2
lim 

2
x  1
x 1
Discuss the behavior of
x2  2
f ( x) 
3( x  2) 2
as x 2
1.9
1.99
1.999
2
2.001
2.01
2.1
187
19867
1998667
?
2001334
20134
214
• As x goes closer to 2 from
the left, f(x) is bigger and
bigger
• As x goes closer to 2 from
the right, f(x) is bigger and
bigger
x2  2
lim

2
x  2 3( x  2)
x2  2
lim

2
x  2  3( x  2)
x2  2
lim

x  2 3( x  2) 2
Theorem 1
Theorem 2
Theorem 3
2) f ( x)  4 x 7  3x 3  x 2
lim (4 x 7  3x 3  x 2 )  lim 4 x 7  
x 
x 
Theorem 4
See examples next page
Horizontal
asymptote is
y=0
Horizontal
asymptote is
y = -3/4
There is no
horizontal
asymptote
Horizontal
asymptote is
y = -1