VERTICAL AND HORIZONTAL ASYMPTOTES

Limits to Infinity and Beyond

I. Theorems:

A.)

B.)

1lim 0

nx x

limx

k k

1 2C.) If lim ( ) and lim ( ) the

sum, difference, constant, and power properties

all apply!!

x xf x L g x L

D.)

providing the root exists.

lim ( ) lim ( )n nx x

f x f x

II. Vertical and Horizontal Asymptotes

A.) Def: The line x = a is a vertical asymptote of the graph of the function f iff

B.) Def: The line y = b is a horizontal asymptote of the graph of the function f iff

lim ( ) x

f x b

lim ( ) or lim ( )x ax a

f x f x

C.) Examples - Find the vertical and horizontal asymptotes for each of the following and describe the behavior at each vertical asymptote.

2

2

2 11.) ( )3

xf xx

32.) ( )3

xf xx

- V.A. – None- H.A. y = 2 Why?

2

2 2

2

2 2

2 1

3lim ( )x

xx xxx x

f x

2

2

2 1( )3

xf xx

2

2

1 12 2 2 0

23 3 1 01 1

limx

x

x

- V.A. – x = -3

- H.A. – y = 1

3( )3

xf xx

3

3

lim

lim

3

3

3 3 603

3 3 603

x

x

3

3

3

3

lim

lim

33

1 1 0 11 01

x

x

xx xxx x

x

x

xx

D.) Example – Evaluate the following limit:

2 40lim

35x

x

x

2 40lim

35x

x

x

2

2 2 240

lim lim35 35

1

1 0lim

0

401

11

x x

x

xx xxx x x

x

III. Sandwich Theorem

GRAPHICALLY

A.) If ( ) ( ) ( ) for all in an open interval

containing the point (with the possible exception

at ) and lim ( ) lim ( ), then lim ( )x c x c x c

g x f x h x x

x c

x c g x L h x f x L

( )f x

( )h x

( )g x

B.) Example -

What do you know about the sin function?

2 2

0

1lim sinxx

x

11 sin 1

x

2 10 sin 1

x

2 2 2 210 sin 1x x x

x

2 2

0

1lim sin 0xx

x

2 2 210 sinx x

x 2 2 2

0 0 0

1lim 0 lim sin limx x x

x xx

2 2

0

10 lim sin 0

xx

x

C.) Example - 20

1lim cosxx

x

2

11 cos 1

x

2

11 cos 1x x x

x

2

1cosx x x

x

20 0 0

1lim lim cos limx x x

x x xx

20

10 lim cos 0

xx

x

20

1lim cos 0xx

x

IV. Limit Theorems

0A.) lim cos 1

0B.) lim sin 0

0

sinC.) lim 1

0D.) lim 1

sin

0

cos 1E.) lim 0

V. Patching

In order to make our trigonometric limits look like A-D of II, we may need to “PATCH” the trig expression. After, we apply our limit properties and verify on our calculator.

A) Examples -

0

sin 31.) lim

x

x

x

0

sin 3lim x

x

x

0 0 0

sin 3 sin 3lim lim .lim

3

3 3

3x x x

x x

x xx x

x x

0 0

sinlim .lim 3

0 0

sin 3lim .lim 3

3x x

x

x 0LET 3 ; lim 0

xx

1 3 3

0

sin 32.) lim

5x

x

x

0

sin 3lim

5x

x

x

0 0 0

sin 3 sin 3 3 3 sin 3lim lim . lim

5 5 5

3

3 3 3x x x

x

x

x x x

x xx

3 31

5 5

0

sin 33.) lim

sin 2x

x

x

0

sin 3lim

sin 2x

x

x

0 0

sin 3 3 sin 3 2lim lim

sin

2

2 2 3 sin 2

3

3 2x x

x x

x x

x x x

x x x

3 31 1

2 2

3

0

sin4.) lim

x

x

x 2

0

sinlim sinx

xx

x 0 1 0

0

1 sin 3lim

cos3 sin 2x

x

x x

0

tan 35.) lim

sin 2x

x

x

0

sin 3lim

sin 2 cos3x

x

x x

0

1 sin 3 2 3lim

cos3 sin 23 2x

x x x

x xx x 3

2

0

1 sin 2 1lim

cos3 cos 2 7x

x

x x x

0

sec3 tan 26.) lim

7x

x x

x

0

1lim sec3 tan 2

7xx x

x

0

1 sin 2lim

cos3 cos 2 7x

x

x x x

0

1 sin 2 2 2lim

cos3 cos 2 7 72x

x x

x x xx

V. Change of Variables

A.) Trig Identities – Know Sum and Difference for sin and cos!!!

B.) Sometimes it is helpful to substitute another variable when evaluating trig limits.

2 1lim 1 cosx x x

1Let =

x

1lim lim 0x x x

0

lim 1 2 cos

2 1lim 1 cosx x x

1 0 1 1 1 2 0 cos 0

C.) Evaluate

2

2lim3cosx

x

x

2

2lim3cosx

x

x

02 203cos

2

Let = 2

x

2

lim 0x

0lim

3cos2

1

3sin 3

0

lim3 0 sin

0lim

3 cos cos sin sin2 2