Transcript Calculus 2.2 - University of Houston

2.2 Limits Involving Infinity
North Dakota Sunset
Photo by Vickie Kelly, 2006
Greg Kelly, Hanford High School, Richland, Washington
4
1
f  x 
x
3
2
1
-4
1
lim  0
x  x
-3
-2
-1
0
1
2
3
4
-1
-2
-3
-4
As the denominator gets larger, the value of the fraction
gets smaller.
There is a horizontal asymptote if:
lim f  x   b
x 
or
lim f  x   b
x 

Example 1:
lim
x 
x
x2  1
 lim
x 
x
x2
x
 lim
1
x  x
This number becomes insignificant as
x  .
 There is a horizontal asymptote at 1.

sin x
f  x 
x
Example 2:
Find:
sin x
lim
x 
x
2
1
-12 -10 -8
-6
-4
-2 -10
-2
1  sin x  1
so for x  0 :
2
6
8
10
12
When we graph this
function, the limit appears
to be zero.
1 sin x 1


x
x
x
1
sin x
1
lim
 lim
 lim
x  x
x 
x  x
x
sin x
0  lim
0
x 
x
4
 by the sandwich
theorem:
sin x
lim
0
x 
x

Example 3:
Find:
5 x  sin x
lim
x 
x
 5 x sin x 
lim  

x   x
x 
sin x
lim 5  lim
x 
x 
x
50
5

Infinite Limits:
4
3
1
f  x 
x
2
1
-4
As the denominator approaches
zero, the value of the fraction gets
very large.
-3
-2
-1
0
If the denominator is negative then
the fraction is negative.
2
3
4
-1
-2
-3
-4
If the denominator is positive then the
fraction is positive.
1
vertical
asymptote
at x=0.
1
lim  
x 0 x
1
lim  
x 0 x

Example 4:
1
lim 2  
x 0 x
1
lim 2  
x 0 x
The denominator is positive
in both cases, so the limit is
the same.
1
 lim 2  
x 0 x

End Behavior Models:
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
f  x
lim
1
x  g  x 
a left end behavior model for f if and only if
f  x
lim
1
x  g  x 

Example 7:
f  x  x  e
x
x
x

e
As
,
approaches zero. (The x term dominates.)
 g  x   x becomes a right-end behavior model.
Test of
model
xe
lim
x 
x
x
e x
 1 0  1
 lim1 
x 
x
Our model
is correct.
x
As x , e increases faster than x decreases,
x
therefore e is dominant.
 h  x   e x becomes a left-end behavior model.
x
Test of
Our model
x
x

e
model

lim

1
 0 1  1
lim
is correct.

x

x
x

x 
e
e

Example 7:
f  x  x  e
x
 g  x   x becomes a right-end behavior model.
 h  x   e x becomes a left-end behavior model.
On your calculator, graph:
y1  x
y2  e  x
y3  x  e  x
Use:
10  x  10
1  y  9

Example 7:
2 x5  x 4  x 2  1
f  x 
3x 2  5 x  7
Right-end behavior models give us:
2 x5 2 x3

2
3x
3
dominant terms in numerator and denominator

Often you can just “think through” limits.
1
lim sin  
x 
 x
0
 lim sin x
x 0
0
p