Warmup – No calculator

bx

axsin lim )3

0

x

7x

sin lim )1

0

xx

x

sin lim )2

2

0

xx

4) Find the average speed in ft/sec of a ball modeled by over the time period [2,6] (feet1062 xxy

2.2 Limits Involving Infinity

1f x

x

Make a table using your calculator

x f(x)

1

10

100

1000

Lets push the value ofx towards infinite

1lim 0x x

1f x

x

1lim 0x x

As the denominator gets larger, the value of the fraction gets smaller.

There is a horizontal asymptote if:

limx

f x b

or limx

f x b

2lim

1x

x

x

Example 1:

2limx

x

x

This number becomes insignificant as .x

limx

x

x 1

There is a horizontal asymptote at 1.

sin xf x

x

Example 2:

sinlimx

x

x Find:

When we graph this function, the limit appears to be zero.1 sin 1x

so for :0x 1 sin 1x

x x x

1 sin 1lim lim limx x x

x

x x x

sin0 lim 0

x

x

x

by the sandwich theorem:

sinlim 0x

x

x

Example 3: 5 sinlimx

x x

x

Find:

5 sinlimx

x x

x x

sinlim 5 limx x

x

x

5 0

5

Infinite Limits (Vertical Asymptotes):

1f x

x

0

1limx x

As the denominator approaches zero, the value of the fraction gets very large.

If the denominator is positive then the fraction is positive.

0

1limx x

If the denominator is negative then the fraction is negative.

vertical asymptote at x=0.

Evaluate without a calculator

145

7)()1

2

xx

xxf

Determine Vertical Asymptotes and evaluate each limit

)( d) )()

)(c) )()

limlim

limlim

22

77

xfxfb

xfxfa

xx

xx

Example 4:

20

1limx x

20

1limx x

The denominator is positive in both cases, so the limit is the same.

20

1 limx x

Limits approaching infinite……. aka… HA

Example:

Sketch a function f(x) that has all of the following properties:

3)(

3)(

)(

)(

4)2(

lim

lim

lim

lim

4

4

xf

xf

xf

xf

f

x

x

x

x

The end

p. 71 (1-8, 9-21 odd, 47-49)