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