Limits
Involving
Infinity
2.2
Bell work
Without a calculator, evaluate each limit either intuitively or
with a table of values.
1 . lim
x 
2.
lim
1
3 . lim
x
x  
x 0
1
x
4 . lim
x 0
If a limit approaches
infinity it technically
does not exist.
1
x
1
x

4
Graphically:
f
x 
3
2
1
1
x
-4
-3
-2
-1
0
1
2
3
4
-1
-2
lim
x  
1
0
x
-3
-4
lim
x 
1
0
x
There is a horizontal asymptote at y = b if:
lim f
x 
x  b
or
lim f
x  
x  b
Asking you to find any horizontal asymptotes is the
same as asking you to find lim f  x  .
x 

4
Graphically:
3
f
x 
1
2
1
x
-4
-3
-2
-1
0
1
2
3
4
-1
lim
x 0
1
-2
 
x
lim
-3
x 0
-4
1

x
There is a vertical asymptote at x = a if:
lim f
x a
x 

or
lim f
x a
x 


More examples by graphing:
f
x 
2x
x 1
lim f ( x )  2
x 
horizontal asymptote
at y = 2
lim  f ( x )  
x  1
vertical asymptote
at x = -1
Without graphing tell whether or not the function has a
horizontal asymptote. If yes, state the equation.
f ( x) 
x
x 1
2
lim
x 
x
x 1
2
 lim
x 
x
x
2
 lim
x 
x
x
1
This number becomes insignificant as x   .

There is a horizontal asymptote at y = 1.

Verify graphically:
f ( x) 
x
x 1
2
The other asymptote
at y = -1 can be
found similarly by
finding
lim f ( x )
x  

End-Behavior
Compare the graphs.
f
 x   3x
4
 2 x  3x  5x  6
3
2
and
f
x 
3x
4
When the value of x is
small, there is a notable
difference in y.
End-Behavior
Zoom out and compare the ends of the graphs…
f
 x   3x
4
 2 x  3x  5x  6
3
2
and
f
x 
3x
4
When the value of x is
large, the y-values are
virtually identical.
End-Behavior
In general, g ( x )  a n x is an end behavior m odel for
n
f ( x )  a n x  a n 1 x
n
n 1
 ...  a 0 , a n  0.
This means that the end behavior model for
f
x 
2 x  7 x  x  3x  1
5
4
f
2
x  2x
5
is
Using end-behavior models to find horizontal asymptotes:
Given: f
x 
Ax
m
Bx
n
If m  n , there is N O horizontal asym ptote.
f ( x) 
3x  1
5
If m  n , the horizontal asym ptote is y  0.
f (x) 
1
x
If m  n , the horizontal asym ptote is y 
3x  7 x  1
2
f ( x) 
2x  3
2
A
.
B
; horizontal asym ptote y 
3
2
a.) Find a power function end behavior model.
b.) Identify any horizontal asymptotes.
1. f
x  6x
2. f
x 
3. f
4. f
5. f
x 
x 
x 
2
 2x  7
2 x  x  x  1
3
2
x3
2 x  3x  7
2
4x  x  3
2
x 9
2
3x  2 x  1
3
x 1
Find the horizontal asymptotes without graphing.
f x 
2x  x  x 1
5
4
2
3x  5x  7
2
NO HORIZONTAL ASYMPTOTE
Proof:
2x  x  x 1
5
lim
x 
lim
x 
2x
5
3x
2
4
2
3x  5x  7
2
lim
x 
2
3
x
3


f
x 
sin x
lim
sin x
x 0
x
1
Find:
x
lim
sin x
x 
x
2
1
-12
-10
-8
-6
lim
x 
-4
sin x
-2
0
-1
-2
2
4
6
8
10
appears to be 0
x
T his can be verified by the Sandw ich T he orem .
12
Find:
lim
5 x  sin x
x 
x
 5 x sin x 
lim 


x 
x 
 x
lim 5  lim
x 
x 
50
5
sin x
x
This also means
that there is a
horizontal
asymptote at y = 5.

Find lim
x 0
lim
x 0
lim
x 0
1
2

2

x
1
x
 lim
x 0
1
x
2
The denominator is positive
in both cases, so the limit is
the same.
1
x
2

This also means
that there is a
vertical asymptote
at x = 0.
Remember that technically, this means the limit does not exist.

Often you can just “think through” limits.
0
1
lim sin  
x 
x
0
p
pg 71 #1-16 (calculator)
#29-38, 47-50, 23-28 (no calc)