Chapter 1
Limits and Their Properties
• 1-2 Finding Limits Graphically and
Numerically
• 1-3 Evaluating Limits Analytically
• 1-4 Continuity and One-Sided Limits
• 1-5 infinite Limits
This
will test
the
“Limits”
of your
brain!
Sect. 1.5
Infinite Limits
To infinity and
beyond …
Infinite Limits
•
Sometimes we do not get a finite limit as an answer. This
occurs when we obtain #/0 as a result. Remember from Sect.
1.3, this indicates there will be a vertical asymptote.
x
1.9
1.99
1.999
2.0
y
1
x2
-10
-100

y
(2.1, 10)


(2.3, 3.33)


(2.5, 2)
-1000
2.001
Und
1000
2.01
100
2.1
10



(1.5, -2)
(1.7, -3.33)



(1.9, -10)
x

Infinite Limits


In these cases, our answer is either
,
 ,
or DNE
To determine which, we can make a table.
Example (from Sect. 1.3)
y

x2  4
Calculate lim 2
.
x 0 x  2 x
x
-.5
y
x 4
x2  2x
2





x


3

-.1
-19
-.01
-199
0
Und
.01
201
.1
21
.5
5
x2  4
lim 2
 
x 0 x  2 x
x2  4
lim 2

x 0 x  2 x
x2  4
lim 2
DNE
x 0 x  2 x
Example (from Sect. 1.3)
2 x 2  10 x  12
Calculate lim
. Graph the function to
x2  9
x 3
confirm.
lim f ( x)  
x 3
lim f ( x)  
x  3
lim f ( x) DNE
x 0
You Try…
Calculate lim
x2
lim f ( x)  
x 2
lim f ( x)  
x 2
1
 x  2
2
. Graph the function to
confirm.
lim f ( x) DNE
x 2
Infinite Limits

As we have already discussed, making a table to calculate a
limit can be quite tedious.

Another strategy is to must find the signs of both the left-hand
and right-hand limits. If the signs are in agreement, then our
solution is either
 or  ; however, if they do not agree,
then our solution is DNE.
2x  5
x 2 x  2
Example: Evaluate lim
Using direct substitution we get
DNE
.
9
0
We must consider the left- and right-hand limits:

lim
f (x )   
x 2



lim
f (x )   
x 2

Since the left-hand limit is negative and the right-hand
limit is positive, they are NOT in agreement, so the
solution is DNE.
4
= -∞
x 0 x 2
Example: Evaluate lim
Direct substitution gives us
.4
0
Let’s look at the left- and right-hand limits:

lim
f (x )   
x  0



lim
f (x )   
x  0

Since the left-hand limit is negative and the right-
hand limit is negative, they’re in agreement, so the
solution is
.
You Try…
Determine the limit to the following. You may use a calculator
to verify your answer.
1)
lim
x 2
3
x2  4
2x  3
2) lim
x 1
x 1

DNE
 3x 2  2 x  1 
3) lim 

2

x 0
2x



 2x 1 

 2x  6 
4) lim 

x 3
5)
 
x2  2x  3
DNE
lim 2
x  3 x  6 x  9
Properties of Infinite Limits
Given lim f ( x)   and lim g ( x)  L
x c
x c
Then
• Sum/Difference lim  f ( x)  g ( x)  
nc
• Product
lim  f ( x)  g ( x)  
nc
 g ( x) 
• Quotient lim 
0

n c
 f ( x) 
L0
Examples
1 

lim 1  2   lim 1  lim 12  1    
x 0 
x  x 0  x 0  x
lim x 2  1
x 1
2
x 1
lim



0
x 1
1
1

lim
x 1 x  1
x 1
2
lim 3 cot x  lim 3  lim cot x  3   
x 0
x 0
x 0

Closure
Explain how vertical asymptotes are related
to finding if a limit goes to ∞ or -∞.