Infinite Limits and
Limits to Infinity: Horizontal
and Vertical Asymptotes
Recall…
• The notation
lim f ( x)  
x c
tells us how the
limit fails to exist by denoting the unbounded
behavior of f(x) as x approaches c.
• Infinity is not a number!
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
xc
f ( x)  g ( x)] 
1. Sum or difference: lim[
x c

1
Consider: f ( x)  x 2
lim[ f ( x)  g ( x)] 
x 0

g ( x)  2
lim f ( x ) 

lim g ( x ) 
2
x0
x0
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
xc
f ( x) g ( x)] 
1. Product: if L > 0 lim[
x c

if L < 0 lim[ f ( x) g ( x)]  
x c
1
Consider: f ( x)  x 2
lim[ f ( x) g ( x)] 
x 0

g ( x)  2
lim f ( x)  
x 0
lim g ( x )  2
x 0
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
xc
1. Quotient: lim  g ( x)  
x c  f ( x) 


0
1
Consider: f ( x)  x 2
 g ( x) 
lim 


x 0
 f ( x) 
0
g ( x)  2
lim f ( x)  
x 0
lim g ( x )  2
x 0
Definition - Vertical Asymptotes
• If f(x) approaches infinity (or negative
infinity) as x approaches c from the left or
the right, then the line x = c is a vertical
asymptote of the graph of f.
Determining Infinite Limits
3
f ( x) 
x
4
g ( x)  2
x
2
h( x )   3
x
lim f ( x) 
x 0
lim f ( x) 
x  0
lim g ( x) 
x  0
lim g ( x ) 
x 0
lim h( x ) 
x  0
lim h( x ) 
x0






The pattern…
Is c even or
odd?
p( x)
f ( x)  c , where p( x) is a polynomial
x and c is a positive integer
odd
Sign of
p(x) when
x=c
positive
odd
negative
even
even
lim f ( x)
x 0

lim f ( x)
x 0


positive


negative



Using the pattern…
3 x
lim

x 0
x

x3  2 x  3
lim

2
x 0
x
2x 1
lim

5
x 0
x


3 x
lim

x 0
x

x3  2 x  3
lim

2
x 0
x
2x 1
lim

5
x 0
x


Using the pattern…
3x 6  6
lim

10
x 0
x

x 2  3x
lim
x 0
x2
x3
 lim

x 0
x

3x 6  6
lim

10
x 0
x

x 2  3x
lim
2
x 0
x
x3
 lim

x 0
x

Limits at Infinity
• lim f ( x)  L denotes that as x
x 
increases without bound, the function
value approaches L
• L can have a numerical value, or the
limit can be infinite if f(x) increases
(decreases) without bound as x
increases without bound
Horizontal Asymptotes
• The line y = L is a horizontal asymptote of f if
lim f ( x)  L
x 
or
lim f ( x)  L
x 
• Notice that a function can have at most two
HORIZONTAL asymptotes (Why?)
4
2
-5
lim f ( x)  ______
0
5
-2
x 
lim f ( x)  ______
0
x 
-4
Horizontal Asymptote(s):__________
4
2
-5
lim f ( x)  ______
2
5
-2
x 
lim f ( x)  ______
2
x 
-4
Note: It IS possible for a graph to cross
its horizontal asymptote!!!!!!
Horizontal Asymptote(s):__________
4
2
-5
lim f ( x)  ______
0
x 
5
-2
lim f ( x)  ______
1
x 
Horizontal Asymptote(s):__________
-4
4
2
-10
-5
5
10
-2
lim f ( x)  ______
0
x 
lim f ( x)  ______
0
x 
Horizontal Asymptote(s):__________
Theorem – Limits at Infinity
1. If r is a positive rational number and c is
any real number, then
c
lim r 
x  x
0
c
lim r 
x  x
0
The second limit is valid only if xr is
defined when x < 0
lim e  0
x 
x
lim e
x 
x

0
Using the Theorem
5
lim 13 
x  x
5
lim  0
x  x
5
5
lim  2  lim  lim 2 
x  x
x  x
x 
lim e
x 
x
0
0
2
lim 2e  lim 2  lim e  0
x  x 
x
x 
x
Guidelines for Finding Limits at
±∞ of Rational Functions
less than
1. If the degree of the numerator is ___________
the degree of the denominator, then the limit of
the rational function is ___.
0
2. If the degree of the numerator is equal
_______
to the
degree of the denominator, then the limit of the
the ratio of the
rational function is the __________________
leading coefficients
_______________________.
than
3. If the degree of the numerator is greater
___________
the degree of the denominator, then the limit of
is infinite
the rational function _______________.
Using the Guidelines…
x3  2 x  3
lim 4
 0
2
x  x  3 x  x
6 x8  12 x  17
lim

x  18 x8  13 x 2  24
1
3
2 x3  9 x  3
lim 3

x  x  7 x 2  1
2
x5  5 x  6
lim 3

x  x  2 x 2  8
∞