Transcript 5ach_15_limits_at_infinity

3.5 Limits Involving Infinity
North Dakota Sunset
What you’ll learn about
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Finite Limits as x→±∞
Sandwich Theorem Revisited
Infinite Limits as x→a
End Behavior Models
Seeing Limits as x→±∞
…and why
Limits can be used to describe the behavior of
functions for numbers large in absolute value.
Finite limits as x→±∞
The symbol for infinity (∞) does not represent a real number.
We use ∞ to describe the behavior of a function when the
values in its domain or range outgrow all finite bounds.
For example, when we say “the limit of f as x approaches
infinity” we mean the limit of f as x moves increasingly far to
the right on the number line.
When we say “the limit of f as x approaches negative
infinity (- ∞)” we mean the limit of f as x moves increasingly
far to the left on the number line.
Horizontal Asymptote
The line y  b is a horizontal asymptote of the graph of a function
y  f  x  if either
lim f  x   b
x 
or
lim f  x   b
x 
Example Horizontal Asymptote
Use a graph and tables to find  a  lim f  x  and
x 
 c  Identify all horizontal asymptotes.
f  x 
f  x .
 b  xlim

x 1
x
f  x  1
 a  lim
x 
f  x  1
 b  xlim

 c  Identify all horizontal asymptotes.
[-6,6] by [-5,5]
y 1
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 

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.

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

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

Often you can just “think through” limits.
1
lim sin  
x 
x
0
 lim sin x
x 0
0
p
Quick Quiz
You may use a graphing calculator to solve the following problems.
1.
A
 B
C
D
E
x2  x  6
Find lim
if it exists
x 3
x 3
1
1
2
5
does not exist
Slide 2- 12
Quick Quiz
2.
A
 B
C
 D
E
3 x  1,

Find lim f  x  =  5
x2
 x  1 ,
5
3
13
3
7

does not exist
x2
x2
if it exists
Slide 2- 13
Quick Quiz
2.
A
 B
C
 D
E
3 x  1,

Find lim f  x  =  5
x2
 x  1 ,
5
3
13
3
7

does not exist
x2
x2
if it exists
Slide 2- 14
Quick Quiz
3.
Which of the following lines is a horizontal asymptote for
3x3  x 2  x  7
f  x 
2 x3  4 x  5
3
A
y

x
 
2
 B y  0
C y 
 D
E
2
3
7
5
3
y
2
y
Slide 2- 15
Quick Quiz
3.
Which of the following lines is a horizontal asymptote for
3x3  x 2  x  7
f  x 
2 x3  4 x  5
3
A y  x
2
 B y  0
2
3
C
y
 D
7
5
3
y
2
E
y
Slide 2- 16
Practice
TEXT
•p. 88, #9 – 52;
•p.89, # 59, 62, 63
•p. 205, #3 – 8, 15 – 20, 21 – 33 odds,
86 – 88.
•
p