2.2 Limits Involving Infinity Hoh Rainforest, Olympic National Park, WA.

2.2 Limits Involving Infinity

Hoh Rainforest, Olympic National Park, WA

Limits & Infinity

• Considerx

xf1

)(

What happens as x gets to be a big number?

Bigxf

1)( Smallxf )(

What happens as x gets even bigger?

BigVeryxf

1)( SmallVeryxf )(

Limits & Infinity

• Considerx

xf1

)(

What if x is wicked huge?

hugewickedxf

1)( SmallWickedxf )(

What if x equals infinity?

1)(xf 0)( xf

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

Horizontal Asymptotes

2lim

1x

x

x 2limx

x

x

This number becomes insignificant as .x

limx

x

x 1

There is a horizontal asymptote at 1.

Horizontal Asymptotes

12lim

x

xx

Horizontal Asymptotes

25

3lim

2

2

x

xx

74

7lim

2

x

xx

sin xf x

x

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

Horizontal Asymptotes

5 sinlimx

x x

x

Find:

5 sinlimx

x x

x x

sinlim 5 limx x

x

x

5 0

5

Horizontal Asymptotes

Limits & Infinity

• Considerx

xf1

)(

What if x is wicked small?

smallwickedxf

1)( HugeWickedxf )(

What if x is an infinitely small positive number?

0

1)( xf )(xf

Whenever a denominator of a fraction approaches zero, the fraction equals +/ - infinity

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.

Vertical Asymptotes:

20

1limx x

20

1limx x

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

20

1 limx x

Vertical Asymptotes:

Is this really true?DNE

Find the Vertical Asymptotes:

342

1)(

2

xx

xxf

Vertical @ x= 2, 3

To find whether the asymptote goes up or down, you must examine the limits.

)(lim2

xfx

)(lim2

xfx

Find the Vertical Asymptotes:

342

1)(

2

xx

xxf

Vertical @ x= 2, 3

To find whether the asymptote goes up or down, you must examine the limits.

)(lim3

xfx

)(lim3

xfx

Find the Vertical & Horizontal Asymptotes:

43

56)(

2

2

xx

xxxf

Find the Vertical & Horizontal Asymptotes:

6

65)(

2

2

xx

xxxf

End behavior models model the behavior of a function as x approaches infinity or negative infinity.

A function g is:

a right end behavior model for f if and only if

lim 1x

f x

g x

a left end behavior model for f if and only if

lim 1x

f x

g x

End Behavior Models:

End Behavior Models:• Find an end behavior model for

252 xxy

End Behavior Models:• Show that y = 3x4 is an end behavior model for • y = 3x4 - 2x2 – 7x +8.

End Behavior Models:• Show that y = 2 is an end behavior model for

21

1

xy

Test ofmodel

Our modelis correct.

xf x x e End Behavior Models

As , approaches zero.x xe(The x term dominates.)

g x x becomes a right-end behavior model.

limx

x

x e

x

lim1

x

x

e

x

1 0 1

xh x e becomes a left-end behavior model.

limx

xx

x e

e

lim 1xx

x

e 0 1 1

As , increases faster than x decreases,x xe

therefore is dominant.xe

Test ofmodel

Our modelis correct.

xf x x e

g x x becomes a right-end behavior model.

xh x e becomes a left-end behavior model.

On your calculator, graph:

1

2

3

x

x

y x

y e

y x e

10 10x

1 9y

Use:

5 4 2

2

2 1

3 5 7

x x xf x

x x

Right-end behavior models give us:

5

2

2

3

x

x

32

3

x

dominant terms in numerator and denominator

Often you can just “think through” limits.

1lim sinx x

0

0lim sinx

x

0