Download Properties of Infinite Limits

1.5
Infinite Limits
Objectives
Determine infinite limits from the left and
from the right.
Find and sketch the vertical asymptotes of
the graph of a function.
Infinite Limits
Graphically
5
lim
 
x 3 x  3
5
lim

x 3 x  3
Infinite Limits
Analytically:
5
lim
 
x 3 x  3
5
lim

x 3 x  3
Plug in number.
If you get # / 0, you know
it’s either ∞ or -∞.
Check sign by plugging in
a number close on the
appropriate side.
Infinite Limits
If the function increases without bound,
the limit is +∞.
If the function decreases without bound,
the limit is -∞.
Example
3
f ( x) 
x4
3
lim
 
x 4 x  4
3
lim
 
x 4 x  4
3
So, lim
DNE
x 4 x  4
Example
f ( x) 
lim
x 3
lim
x 3
2
 x  3
2
 x  3
2
 
2
 
2
 x  3
So, lim
x 3
2
2
 x  3
2
 
Examples
1
1
1
lim
  and lim
  so lim
DNE
x 1 x  1
x 1 x  1
x 1 x -1
1
lim
 
2
x 1 ( x  1)
1
1
1
lim
  and lim
  so lim
DNE
x 1 x  1
x 1 x  1
x 1 x  1
1
lim
 
2
x 1 ( x  1)
As x approaches 1, the
graphs become
arbitrarily close to the
vertical line x=1.
This line is called a
vertical asymptote.
If f(x) approaches ∞ or -∞, as x approaches c
from the right or from the left, then the line x=c
is a vertical asymptote of the graph of f.
Theorem 1.14
Let f and g be continuous on an open
interval containing c. If f(c)≠0, g(c)=0 and
there exists an open interval containing c
such that g(x)≠0 for all x≠c in the interval,
then the graph of the function given by
h(x)=f(x) / g(x) has a vertical asymptote at
x=c.
(Vertical asymptotes occur at numbers that
make the denominator 0, but NOT the
numerator).
Vertical Asymptotes
Find all the vertical asymptotes:
1
f ( x) 
x2
( x  1)( x  2)
f ( x) 
( x  2)( x  3)
x=2
x=3
x 2  2 x  8 ( x  4)( x  2)
f ( x) 

2
x 4
( x  2)( x  2)
x= -2
Theorem 1.15
Properties of Infinite Limits
Let c and L be real numbers and let f and g be
functions such that lim f ( x)   and lim g ( x)  L.
x c
1.sum/difference:
2.product:
x c
lim[ f ( x)  g ( x)]  
x c
lim[ f ( x) g ( x)]  , L  0
x c
lim[ f ( x) g ( x)]  , L  0
x c
3.quotient:
lim g ( x)
x c
f ( x)
0
What do you think?
∞+∞=∞
-∞ - ∞ = -∞
#/∞=0
∞ - ∞ = ???
Example
1 

lim 1  2   1    
x 0
 x 
Example
lim(
x

1)

2

2
x 1
lim(cot

x
)



x 1
x 1
2
lim

0
x 1 cot  x

2
Example
lim 3  3
x 0
lim (cot x)  
x 0
lim 3cot x  3()  
x 0
Homework
1.5 (page 88)
#5, 7, 13-57 odd
(Don’t graph)
Handout (2.5)
#11-19 odd
#39, 47, 51