Infinite Limits
Lesson 1.5
Infinite Limits
Two Types of infinite limits.
Either the limit equals infinity or the limit is approaching infinity.
We are going to take a look at when the limit equals infinity, for now.
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)
The function f(x) will have a vertical asymptote at x = a if we obtain any of
the following limits:
)(lim xfax
)(lim xfax
)(lim xfax
Definition of Infinite Limits
M --------------
f(x) increases without bound as x c
NOTE: may decrease without bound ie: go to negative infinity!!
Vertical Asymptotes
• When f(x) approachesinfinity as x → c– Note calculator often
draws false asymptote• Vertical asymptotes generated by
rational functions when g (x) = 0
c
( )( )( )f xh xg x
Theorem 1.14Finding Vertical Asymptotes
• If the denominator = 0 at x = c AND the numerator is NOT zero, we have a vertical asymptote at x = c!!!!!!! IMPORTANT
• What happens when both num and den are BOTH Zero?!?!
A Rational Function with Common Factors(Should be x approaching 2)
• When both numerator and denominator are both zero then we get an indeterminate form and we have to do something else …
– 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.
2
22
2 8lim4x
x xx
2
4lim , 22x
x xx
A Rational Function with Common Factors, cont….
• Direct sub yields 0/0 or indeterminate form. When we simplify eliminate indeterminate form and we learn that there is a vertical asymptote at x = -2 by theorem 1.14.
• Take lim as x-2 from left and right
• 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
2
22
2 8lim4x
x xx
2
22
2 8lim4x
x xx
Determining Infinite Limits
• Denominator = 0 when x = 1 AND the numerator is NOT zero– Thus, we have vertical
asymptote at x=1• But 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!
2 2
1 1
3 3 lim and lim1 1x x
x x x xFindx x
Infinite Limits:
1f xx
0
1limx x
As the denominator approaches zero, the value of the fraction gets very large.
If the denominator is positive then the fraction is positive.
0
1limx x
If the denominator is negative then the fraction is negative.
vertical asymptote at x=0.
Example 4:
20
1limx x
20
1limx x
The denominator is positive in both cases, so the limit is the same.
20
1 limx x
Properties of Infinite Limits• Given
Then• Sum/Difference
• Product
• Quotient
lim ( ) and lim ( )x c x cf x g x L
lim ( ) ( )x c
f x g x
lim ( ) ( ) 0x c
f x g x L
( )lim 0( )x c
g xf x
lim ( ) ( ) 0x c
f x g x L
Find each limit, if it exists.
4
11. lim4x x
6
4
2
-2
-4
-6
-5 5
Find each limit, if it exists.
4
11. lim4x x
13.999 4
1VS
Very small negative #
One-sided limits will always exist!
6
4
2
-2
-4
-6
-5 5
1
12. lim1x x
6
4
2
-2
-4
-6
-5 5
1
12. lim1x x
10.999 1
1VS
This time we only care if the two sides come together—and where.
6
4
2
-2
-4
-6
-5 5
1
1lim1x x
1
1lim1x x
11.001 1
1VS
DNE
Can’t do Direct Sub, need to go to our LAST resort…
check the limits from each side.
3. Find any vertical asymptotes of2
2
2 8( )4
x xf xx
6
4
2
-2
-4
-6
-5 5
3. Find any vertical asymptotes of2
2
2 8( )4
x xf xx
Discontinuous at x = 2 and -2.
4 22 2
x xx x
42
xx
V.A. at x = -2 3
2Hole at 2,
6
4
2
-2
-4
-6
-5 5
Try It Out
• Find vertical asymptote
• Find the limit
• Determine the one sided limit
2
2( )1xg x
x x
2
24lim
16x
xx
3
2 1
1( ) lim ( )1 x
xf x f xx x
Methods
• Visually: Graphing• Analytically: Make a table close to “a”• Substitution: Substitute “a” for x
If Substitution leads to:1) A number L, then L is
the limit
2) 0/k, then the limit is
zero
3) k/0, then the limit is ±∞, or
dne
4) 0/0, an indeterminant form, you must do more!
