### 1.5 Notes - Infinite Limits and 3.5

```1.5 Notes - Infinite Limits
and 3.5 Notes - Limits at
Infinity
AP Calculus AB
Created by Ms. Hernandez
Modified by Mr. Harp
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
Part I: Infinite Limits:
4
3
1
f  x 
x
2
1
-4
-3
-2
-1
0
1
2
3
4
-1
1
lim  
x 0 x
-2
-3
-4
vertical
asymptote
at x=0.
1
lim  
x 0 x

online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc02_2.ppt
IMPORTANT NOTE:
f ( x)  
x c
does NOT mean the limit exists!
“On the contrary, it tells HOW
the limit FAILS to exist.”
Page 83 Dr. Larson & Dr. Hostetler
www.vcsc.k12.in.us/lincoln/math/CaseNotes/.../InfiniteLimits.ppt
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.
www.vcsc.k12.in.us/lincoln/math/CaseNotes/.../InfiniteLimits.ppt
Digging deeper…
 Infinity is a very special idea. We know
we can't reach it, but we can still try to
work out the value of functions that
have infinity in them.
http://www.mathsisfun.com/calculus/limits-infinity.html
 Question: What is the value of 1/∞ ?
 Maybe we could say that 1/∞ = 0, ... but
if we divide 1 into infinite pieces and they
end up 0 each, what happened to the 1?
 In fact 1/∞ is known to be undefined.
http://www.mathsisfun.com/calculus/limits-infinity.html
But We Can Approach It!
x
1/x
1 1.00000
2
4
10
100
0.50000
0.25000
0.10000
0.01000
1,000 0.00100
10,000 0.00010
http://www.mathsisfun.com/calculus/limits-infinity.html
The limit of 1/x as x approaches
Infinity is 0
Furthermore:
1
lim n  0, n  0
x  x
http://www.mathsisfun.com/calculus/limits-infinity.html
Limits at Infinity
 x2 1 

lim 2
x  x  1




 lim
x 


 Divide through by the
x2 1 
 2
2
x
x 
x2 1 
 2
2
x
x 
1

1 2
x
 lim
x 
1
1 2
x

highest power of x
 Simplify





1 0
 
1 0
1
 Substitute 0 for 1/xn
www.mrsantowski.com/MCB4U/Notes/PowerPointNotes/BCC016.ppt
Example
3x 2  5 x  1
lim
x
2  4 x2
Divide
by x2
3 5  1 2
x
x
 lim
2 4
x 
x2

 x   lim  1 x 
lim 3  lim 5
x 
x 
 x   lim 4
lim 2
x 
x 
2
2
x 
3 0 0
3


04
4
www.rowan.edu/open/depts/math/.../Limits%20and%20Continuity.pp.
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 85
 Denominator = 0 at x = c AND the
numerator is NOT zero
 Thus,
 What
we have vertical asymptote at x = c
happens when both numerator
and denominator are BOTH Zero?!?!
A Rational Function with Common
Factors

When both the num. and den. are both zero
then we get an indeterminate form and we
have to do something else …
 Ex 3 pg 86
x2  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 find 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 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
x  2x  8
lim

2
x 2
x 4
2

x2  2 x  8
lim
 
2
x 2
x 4
Take values close to –2 from the right and values
close to –2 from the left … Make a table and you will
see values approaching 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 (Finite#/infinity = 0)
 Same properties for lim f ( x )  
x c
 Ex 5 pg 87
Definition of a Horizontal Asymptote
If f(x) approaches a finite limit L as x
approaches infinity or negeative
infinity, then y = L is a horizontal
asymptote of f.
www.vcsc.k12.in.us/lincoln/math/CaseNotes/.../InfiniteLimits.ppt
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
lim
f
(
x
)

(d)
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
2 x 1
1

lim f ( x)  lim
 lim
 lim  2    2
x 
x 
x 
x 
x
x
x

1/big = little and 1/little = big
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

y=2 and y=–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 

3.5 Limit at Infinity
 Horizontal
asymptotes!
 Finite#/infinity = 0
 Infinity/infinity
 Divide
the numerator & denominator by a
denominator degree of x
Some examples
 Ex
2-3 on pages #199-200
 What’s the graph look like on Ex 3.c
 Called
 KNOW
oblique asymptotes (not in calc AB)
Guidelines on page 201
example:
8aPvXI
2 horizontal asymptotes
 Ex
4 pg 202
 Is the method for solving lim of f(x) with
2 horizontal asymptotes any different
than if the f(x) only had 1 horizontal
asymptotes?
Limits involving Trig functions
 Ex
5 pg 203
 What is the difference in the behaviors
of the two trig functions in this example?
 Oscillating toward no value vs
oscillating toward a value
Word Problems !!!!!
 Taking
information from a word problem
and apply properties of limits at infinity
to solve
 Ex 6 pg 203
A word on infinite limits at infinity
 Sometimes,
when we determine a limit
of a f(x) as x->∞, it comes out ∞ or - ∞
 Ex 7 on page 204
 Uses
 Ex
property of f(x)
8 on page 204
 Uses
LONG division of polynomials-Yuck!
The shortcut for limits AT infinity
Whenever dealing with evaluating limits of the forms
()
()
lim
, or lim
,
→∞ ()
→−∞ ()
where P(x) and Q(x) are both polynomials,
you can ignore all but the highest degree term in the
numerator and the highest degree term in the
denominator! This is because as x approaches ∞ or
-∞, the highest degree term dominates.
The next 5 slides go through this in detail.
Limits at Infinity of
Polynomial Functions
What about limits at infinity for polynomial functions?
As x increases without bound in either the positive or the
negative direction, the behavior of the polynomial graph will
be determined by the behavior of the leading term (the
highest degree term). The leading term will either become
very large in the positive sense or in the negative sense
(assuming that the polynomial has degree at least 1). In the
first case the function will approach  and in the second
case the function will approach -.
In mathematical shorthand, we write this as lim f ( x )  
x  
This covers all possibilities.
31
Limits at Infinity and
Horizontal Asymptotes
A line y = b is a horizontal asymptote for the graph of y = f (x)
if f (x) approaches b as either x increases without bound or
decreases without bound. Symbolically, y = b is a horizontal
asymptote if
lim f ( x)  b or
x  
lim f ( x)  b
x 
In the first case, the graph of f will be close to the horizontal
line y = b for large (in absolute value) negative x.
In the second case, the graph will be close to the horizontal line
y = b for large positive x.
Note: It is enough if one of these conditions is satisfied, but
frequently they both are.
32
Example
This figure shows the graph of a function with two
horizontal asymptotes, y = 1 and y = -1.
33
Horizontal Asymptotes
SPECIFICALLY for Rational Functions
If
am x m  am1 x m1    a1 x  a0
f ( x) 
, am  0, bn  0
n
n 1
bn x  bn1 x    b1 x  b0
am x m
lim f ( x)  lim
then x
x  b x n
n
There are three possible cases for these limits.
1. If m < n, then lim f ( x)  0
x  
The line y = 0 (x axis) is a horizontal asymptote for f (x).
am
f ( x) 
2. If m = n, then xlim

bm
The line y = am/bn is a horizontal asymptote for f (x) .
3. If m > n, f (x) does not have a horizontal asymptote.
34
Horizontal Asymptotes
of Rational Functions (continued)
Notice that in cases 1 and 2 on the previous slide that the limit is
the same if x approaches  or -. Thus a rational function can
have at most one horizontal asymptote. (See figure). Notice
that the numerator and denominator have the same degree in this
example, so the horizontal asymptote is the ratio of the leading
coefficients of the numerator and denominator.
y
3x  5 x  9
2x2  7
2