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 as x 11
1)(
x
xf
.9 .99 .999 .9999 1 1.0001 1.001 1.01 1.1
-10 -100 -1000 -10000 ? 10,000 1000 100 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
existnotdoesx
x
x
x
x
x
__1
1lim
1
1lim
1
1lim
1
1
1
Describe the behavior of at 1 and -11
2)(
2
2
x
xxxf
.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
because
1
2lim
1
2lim
2
2
1
2
2
1
x
xx
x
xx
x
x
Discuss the behavior of as x 22
2
)2(3
2)(
x
xxf
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
2
2
2
2
2
2
2
2
2
)2(3
2lim
)2(3
2lim
)2(3
2lim
x
x
x
x
x
x
x
x
x
Theorem 1
Theorem 2
Theorem 3
7237
237
4lim)34(lim
34)()2
xxxx
xxxxf
xx
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
