VERTICAL AND HORIZONTAL ASYMPTOTES Limits to Infinity and Beyond
Jan 03, 2016
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
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
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
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
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