Limits involving infinity
I] Infinity, Minus Infinity as a limit as x approaches a:
The line x=a is a vertical asymptote of the function:
When lim f ( x ), lim f ( x ), or lim f ( x ) is plus or minus infinity then the function f(x) has
x a
x a

x a

the line x=a as a vertical asymptote.
Basic limits to remember:
1
lim 
x 0
 ,
lim 
x 0
x
lim ln | x |  ,
x 0
1
  ,
x 0
is a vertical asymptote
x
x  0 is is a vertical asymptote
f (x) 
of f ( x )  ln | x |
x
2
 3 x  10
x
lim f ( x )
x 2
lim
x  2
2
4
.
This is a rational function and substitution gives the indeterminate form 0/0.
We factor numerator and denominator.
b)
1
x
Example: Find each limit for the function
a)
of f ( x ) 
f ( x )  lim
x5
DNE
x  2 x  2
( x  5 )( x  2 )
( x  2 )( x  2 )

x5
x2

7
as
4
x  2.
because substitution gives 3/0. In fact the right side
approaches positive infinity, the left side approaches minus infinity. The line x= -2 is a
vertical asymptote.
When both sides go to positive infinity, we say the limit is infinity. Similarly if both sides
go to negative infinity, the limit is minus infinity.
For example, we write
Example: Find
lim
x0
lim
1
x 0
ln | x |
x
2

since both sides approach positive infinity.
or show it does not exist. You can graph the function and see
x
its behavior near 0.
lim
x 0
ln | x |

x
 
lim
x 0
ln | x |

x

and
lim
x0
ln | x |
x
does not exist.
II] Limits as x approaches positive infinity, and as x approaches minus infinity:
Horizontal Asymptotes:
When a function approaches one number, L, as x goes to infinity or to minus infinity, we
say f(x) has horizontal asymptote y=L. There can be horizontal asymptotes if the function
approaches one number as x goes to minus infinity and a different number as x goes to
positive infinity. We will see functions that have horizontal asymptotes but first lets look
at polynomials.
Polynomials never have horizontal asymptotes and always approach infinity or minus
infinity as x goes to infinity or to minus infinity.
The tendency of a polynomial as x approaches plus or minus infinity is the same as
the tendency of its leading term. You can see this two ways:
Graph Y 1  x 3  5 x 2  3 x  10 Y 2  x 3 in your calculator. Use a window in which Xmin
and Xmax are large, say Xmin=1000 and Xmax=1100. The graphs are nearly identical.
If x=1000, then the leading term is on the order of 1000 times any other term.
lim x
3
x 
lim
x  
x

3
 
5
x  
The same works at minus infinity:
x 
lim Y 1  
so
lim (  2 x
Ex.
lim Y 1  
so
x  
4
 20 x
 15 x
3
 35 ) 
lim  2 x
5

x  
. Any odd power of x will approach
minus infinity as x approaches minus infinity because an odd power of a negative is
negative. Then the coefficient -2 makes it positive.
Rational functions: You only need to consider the tendency of the ratio of the leading
terms. This only works for ratios.
Example:
2x
lim
x 
Example:
lim
x 
4
6x
3
 15 x  16 x
4
2
3
 32 x  50 x
3x
2
 lim
x 
 4 x  10
3
2 x  5x
2
2x
 10 x  15
6x
4
x 

6
1
3
2
3x
 lim
2

4
2x
 lim
3
y=1/3 is a horizontal asymptote.
3
x  2x
0
y=0 is a horizontal
asymptote.
Example:
4x
lim
4
 3x
2
3
9
5x  2 x
x  
2

4x
lim
x  
5x
4
3

lim
x  
4x
 
5
There is no horizontal
asymptote.
Other limits: "infinity minus infinity" is indeterminate.
Example:
Find

lim 
x
x 
2
 4x 
x
2

1

Both terms tend to infinity so we have
"infinity minus infinity".
x
2
 4x 
x
2
1 
( x
2
 4x 
x
x
4x 1
x
2
 4x 

x
2
1
4x
2x
 2
2
2
 1 )(
 4x 
x
2
x
 4x 
2
1
x
2
 1)

(x
2
x
2
 4x)  (x
 4x 
2
x
2
 1)
1

Exercises: Show
lim ( x
x 
2
 6x  x)  3
and
lim ( x
x  
2
 6 x  x )  3
Show that every odd degree polynomial has at least one root. Use the Intermediate Value
Theorem.