BC 1
Limits 3
Name:
Limits Involving Infinity
Let   x  
4 x2  2 x
.
x2  x  6
How do we find vertical asymptotes of a function? What are the vertical asymptotes of ƒ?
Investigate the behavior of  around these asymptotes by find each of the following limits.
lim    x  
lim   x  
x2
x3
lim   x  
x2
lim   x  
x3
How do we find horizontal asymptotes of a function?
The horizontal asymptote of  is
.
Graph the function to confirm your limits and your guess for the asymptote.
For each of the following, think about what happens to the fraction as x grows larger and
guess the value of each limit. Note that this is another way of asking you to find k where y = k is
the horizontal asymptote. Graph each function to check your guess.
(1)
(3)
2x 1
x 
x
lim
lim
(2)
5 x3
x 2 x
2
(4)
1
x 3x 2
lim
What happens to y as x  –?
BC 1
Lim 3.1
 2x
2  4x
x x3  5 x
To approach this more methodically, consider first y 
What happens to y as x  +?
4 x2  1
lim
1
xn
where n  1.
Return to   x  
4 x2  2 x
x2  x  6
. Again, algebra will help to change the form of the function.
Multiply both the numerator and denominator by
1
x
2
since x2 is the largest power of x in the
 4x  2x      lim 4  2x
x x 2  x  6

  x1  x 1  1x  x6
2
1
x2
denominator. We have lim
2
2
6
x2
 4 , since the terms
all approach 0 as x approaches  .
Determine the following limits.
(1)
(2)
(3)
(4)
4 x3  5
lim
x x3
lim
4 x4  6 x  1
1  2 x4
x
lim
 2 x2
4 x8  6 x3  5 x
x
lim
x10  1
1  4 x2
x 3x3
 6x
Assorted: Determine each of the following limits.
x2
x2
lim
lim
(5a)
(b)
x2 x  2
x2 x  2
(6)
(8)
lim 
x1
(7)
lim
x3
lim
x2
x2
x2
x2
x3
3x  2, x  1
Given   x   
, find:
 4 x  1, x  1
(a)
BC 1
x2
x  x  1
(c)
lim   x 
x 1
(b)
lim   x 
x1
Lim 3.2
(c)
lim   x 
x1
2 1
, , and
x x