1.5 Infinite Limits
IB/AP Calculus I
Ms. Hernandez
Modified by Dr. Finney
AP Prep Questions / Warm Up
No Calculator!
ln x
lim
x 1 x
(a) 1 (b) 0 (c) e (d) –e (e) Nonexistent
( x  2)
lim 2
x 2 x  4
(a) –1/4 (b) –1/2 (c) 0 (d) 1 (e) DNE
AP Prep Questions / Warm Up
No Calculator!
ln x ln1 0
lim

 0
x 1 x
1
1
(a) 1 (b) 0 (c) e (d) –e (e) Nonexistent
( x  2)
( x  2)
1
1
lim 2
 lim
 lim

x 2 x  4
x 2 ( x  2)( x  2)
x 2 ( x  2)
4
(a) –1/4 (b) –1/2 (c) 0 (d) 1 (e) DNE
Infinite Limits
 If
function values keep INCREASING
WITHOUT BOUND
________________
as x approaches a given value
INFINITY
we say the limit is _____________.
lim f ( x)  
x c
Infinite Limits
 If
function values keep DECREASING
WITHOUT BOUND
________________
as x approaches a given value
- INFINITY
we say the limit is _____________.
lim f ( x)   
x c
IMPORTANT NOTE:
The equal sign in the statement
lim f ( x)   does NOT mean the
x c
limit exists!
On the contrary, it tells HOW
the limit FAILS to exist.
Examples
1
lim

x2
x2
1
THINK : lim

x2
x  2
1
THINK : lim

x2
x  2
Examples
lim
x2
1
 x  2
THINK : lim
x 2

THINK : lim
x2

2

1
 x  2
2
1
 x  2
2


REMEMBER:
The equal sign in the statement
lim f ( x)   does NOT mean the
x c
limit exists!
On the contrary, it tells HOW
the limit FAILS to exist.
Definition of a Vertical Asymptote
If f(x) approaches infinity or negative infinity
as x approaches c from the left or right,
then x = c is a vertical asymptote of f.
VA @ x  c if lim f ( x)   or lim f ( x)  
x c
x c
1.5 Infinite Limits
 Vertical
asymptotes at x=c will give you
infinite limits
 Take the limit at x=c and the behavior of
the graph at x=c is a vertical asymptote
then the limit is infinity
 Really the limit does not exist, and that
it fails to exist is b/c of the unbounded
behavior (and we call it infinity)
Determining Infinite Limits from a
Graph
 Example
1 pg 84
 Can you get different infinite limits from
the left or right of a graph?
 How do you find the vertical asymptote?
Finding Vertical Asymptotes
 Ex
2 pg 84
 Denominator = 0 at x = c AND the
numerator is NOT zero
 Thus,
 What
we have vertical asymptote at x = c
happens when both num and den
are BOTH Zero?!?!
A Rational Function with Common
Factors

When both num and den are both zero then
we get an indeterminate form and we have to
do something else …
2
 Ex 3 pg 86
x  2x  8
lim
x 2


x 4
2
Direct sub yields 0/0 or indeterminate form
We simplify to find vertical asymptotes but how do
we solve the limit? When we simplify we still have
indeterminate form.
x4
lim
, x  2
x 2 x  2
A Rational Function with Common
Factors
 Ex
3 pg 86: Direct sub yields 0/0 or
indeterminate form. When we simplify
we still have indeterminate form and we
learn that there is a vertical asymptote
at x = -2.
 Take lim as x-2 from left and right
2
2
x  2x  8
x  2x  8
lim
lim
2
x 2
x 2
x 4
x2  4
A Rational Function with Common
Factors


Ex 3 pg 83: Direct sub yields 0/0 or indeterminate
form. When we simplify we still have indeterminate
form and we learn that there is a vertical asymptote
at x = -2.
Take lim as x-2 from left and right
x  2x  8
lim

2
x 2
x 4
2

x  2x  8
lim
 
2
x 2
x 4
2
Take values close to –2 from the right and values
close to –2 from the left … Table and you will see
values go to positive or negative infinity
Determining Infinite Limits
 Ex
4 pg 86
 Denominator = 0 when x = 1 AND the
numerator is NOT zero
 Thus,
we have vertical asymptote at x=1
is the limit +infinity or –infinity?
 Let x = small values close to c
 Use your calculator to make sure – but
they are not always your best friend!
 But
Properties of Infinite Limits
 Page
87
lim f ( x)  
x c
lim g ( x)  L
x c
 Sum/difference
 Product
L>0, L<0
 Quotient (#/infinity = 0)
 Same properties for lim f ( x )  
x c
 Ex 5 pg 87
Asymptotes & Limits at Infinity
For the function
(a) lim f ( x )
2x 1
f ( x) 
, find
x
x 
(b) lim f ( x )
x 
(c) lim f ( x )
x 0
(d) lim f ( x )

x 0
(e) All horizontal asymptotes
(f) All vertical asymptotes
Asymptotes & Limits at Infinity
2x 1
f ( x) 
x
For x>0, |x|=x (or my x-values are positive)
2x 1
2x 1
1

lim f ( x)  lim
 lim
 lim  2    2
x 
x 
x 
x 
x
x
x

1/big = little and 1/little = big
sign of denominator leads answer
For x<0 |x|=-x (or my x-values are negative)
2x 1
2x 1
1

lim f ( x)  lim
 lim
 lim  2    2
x 
x 
x 
x 
x
x
x

2 and –2 are HORIZONTAL Asymptotes
Asymptotes & Limits at Infinity
2x 1
f ( x) 
x
2x 1
2x 1
1

lim f ( x)  lim
 lim
 lim  2  
x 0
x 0
x 0
x 0 
x
x
x
1 
1 

2   2
  2      lim DNE
x 
little 

2x 1
2x 1
1

lim f ( x)  lim
 lim
 lim  2    2
x 
x 
x

x 
x
x
x

1 
1 


2



2


 
  2      lim DNE
x 
little 

1.5 Limit at Infinity
 Horizontal
asymptotes!
 Lim as xinfinity of f(x) = horizontal
asymptote
 #/infinity = 0
 Infinity/infinity
 Divide
the numerator & denominator by a
denominator degree of x