Transcript Aim: Do Limits at Infinity make sense?

Aim: Do Limits at Infinity make sense?
Do Now:
List the characteristics of the
following function
2
3x
f  x  2
x 1
Aim: Limits at Infinity
Course: Calculus
Do Now
q x  =
3x 2
4
y =3
x 2+1
3
f(x)  3 as
x  -
-6
-4
f(x)  3 as
x
2
1
-2
2
4
6
8
-1
x decreases w/o bound
x -  -100
f(x) 3 
x increases w/o bound
-10 -1 0 1
10
100

2.9997 2.97 1.5 0 1.5 2.97 2.9997  3
f(x) approaches 3
lim f  x   3
x 
Aim: Limits at Infinity
f(x) approaches 3
lim f  x   3
x 
Course: Calculus
Horizontal Asymptote
The line y = L is a horizontal asymptote of
the graph of f if
lim f  x   L or lim f  x   L
x 
q x  =
x 
3x 2
4
y =3
x 2+1
3
f(x)  3 as
x  -
-6
-4
f(x)  3 as
x
2
1
-2
2
4
6
-1
lim f  x   3
x 
Aim: Limits at Infinity
lim f  x   3
x 
Course: Calculus
8
Limits at Infinity for Rational Functions
For the rational function f(x) = P(x)/Q(x),
where
P(x) = anxn + . . . .a0
and
Q(x) = bmxm + . . . .b0
the limit as x approaches positive or
negative infinity is as follows:
 0, n  m

lim f ( x )   an
x 
,
n

m
b
 m
n is the highest
power of
numerator; m is
highest power of
denominator
If n > m, the limit does not exist.
Aim: Limits at Infinity
Course: Calculus
Definition of Limits at Infinity
If f is a function and L1 and L2 are real
numbers, the statements
lim f ( x )  L1
Limit as x approaches -
x 
lim f ( x )  L2
x
Limit as x approaches 
denote the limits at infinity. The first is
read “the limit of f(x) as x approaches - is
L1” and the second is read “the limit of f(x)
as x approaches  is L2”.
1
If n is a positive real number, then lim n  0
x  x
1
If x < 0, then lim n  0
x  x
Aim: Limits at Infinity
Course: Calculus
Operations with Limits
If r is a positive rational number and c is any
real number, then
c
lim r  0
x  x
Also, if xr is defined when x < 0, then
c
lim r  0
x  x
Find the limit:
2 

lim  5  2 
x 
x 

2
lim 5  lim 2
x 
x  x
5–0=5
Aim: Limits at Infinity
Course: Calculus
Evaluating Limit at Infinity
 2x  1 
Find the limit: lim 

x 
x

1


4
graphically
2
-5
5
-2
from study of rational polynomial functions
If degree of p = degree of q, then the line
y = an/bm is a horizontal asymptote, the
ratio of the coefficients.
Note: a rational function will always
approach the same horizontal asymptote to
Aim: Limits at Infinity
Course: Calculus
the right and left.
Limits at Infinity
x 1
f ( x) 
2x
x+1
f x  =
2x
4
horizontal
asymptote
y = 1/2
2
1
lim f ( x ) 
x 
2
x approaches 
-5
x approaches -
1
lim f ( x ) 
x  
2
5
-2
-4
vertical
asymptote
x=0
What happens to f(x) as x
approaches  infinity?
Aim: Limits at Infinity
Course: Calculus
Model Problem
3
l im(4  2 )
x
x
Find the limit
Algebraically:
3
3
l im(4  2 )  lim4  lim 2
The limit of
x
x 
x  x
x
f(x) as x
1
 lim4  3 lim 2 approaches
x 
x  x
 is 4.
 4  3(0)  4
Graphically:
h x  = 4-
3
x2
4
2
-5
Aim: Limits at Infinity
5
-2
Course: Calculus
Application
You are manufacturing a product that costs $0.50
per unit to produce. Your initial investment is
$5000, which implies that the total cost of
producing x units is C = 0.5x + 5000. The average
cost per unit is
C 0.5 x  5000
C 
x
x
Find the average cost per unit when
a) x = 1000,
0.5(1000)  5000
C
 $5.50
1000
b) x = 10,000,
0.5(10000)  5000
C
 $1.00
10000
c) when x  
limC 
x 
Aim: Limits at Infinity
0.5 x  5000
 $0.50
x
Course: Calculus
Evaluating Limit at Infinity
 2x  1  = 2
Find the limit: lim 

x 
x

1


algebraically:
divide all terms by highest power of x
in denominator
2x 1
1

2
x
lim x x  lim
x  x
1 x 
1

1
x x
x
20

2
1 0
Aim: Limits at Infinity
1
lim 2  lim
x 
x  x

1
lim1  lim
x 
x  x
c
lim r  0
x  x
Course: Calculus
Model Problem
Find the limit for each as x approaches .
 2x  3
f ( x) 
3x2  1
2 3
  2
 2x  3
00
x
x
lim 2
 lim

0
x  3 x  1
x 
1
3 0
3 2
x
 2x2  3
f ( x) 
3x2  1
3
 2 2
 2x2  3
 2 0
2
x
lim
 lim


x  3 x 2  1
x 
1
3 0
3
3 2
x
 2x3  3
A polynomial
f ( x) 
3x2  1
3 tends to behave
 2 x  2 as its highest 2x3  3
x
lim

lim
no limit
x  3 x 2  1
x 
1 powered term
3  2 behaves as x
x
approaches

Aim: Limits at Infinity
Course:
Calculus
Functions with Two Horizontal Asymptotes
Determine limits for each
3x  2
3x  2
lim 
lim 
x 
x 
2 x2  1
2 x2  1
for x = > 0, x =
3x  2
x

2
2x  1
x2
2
x , then divide
2
2
3
3
x 
x
2
1
2x  1
2 2
2
x
x
2
3
30
3
x
lim 


x 
1
20
2
2 2
x
Aim: Limits at Infinity
Course: Calculus
Functions with Two Horizontal Asymptotes
Determine limits for each
3x  2
3x  2
lim 
lim 
x 
x 
2 x2  1
2 x2  1
for x = < 0, x = - x 2 , then divide
3x  2
x

2
2x  1
2
2
3
3
x
x

1
2 x2  1
 2 2

2
2
x
x
 x
2
3
30
3
x
lim 


x 
1
 2 0
2
 2 2
x
Aim: Limits at Infinity
Course: Calculus
Limits Involving Trig Functions
Determine limits for each
sin x
lim  sin x
lim 
x 
x 
x
f 2x  = sin x 
5
10
as x approaches infinity x oscillates
between -1 and 1
conclusion:
a limit does not exist
f(x) oscillates between two
fixed values as x approaches c.
Aim: Limits at Infinity
Course: Calculus
Limits Involving Trig Functions
Determine limits for each
sin x
lim  sin x
lim 
x 
x 
x
1
h x  =
0.5
1
x
5
-0.5
10
g x  =
-1
x
because -1 < sin x < 1, for x > 0,
1 sin x 1
 

x
x
x
1
1
lim    0
lim   0
x 
x 
x
x
sin x
0
Squeeze Theorem: lim 
x 
x
-1
Aim: Limits at Infinity
Course: Calculus
Model Problem 1
f(t) measures the level of oxygen in a pond,
where f(t) = 1 is the normal (unpolluted)
level and the time t is measured in weeks.
When t = 0, organic waste is dumped into the
pond, and as the waste material oxidizes, the
level of oxygen in the pond is
t2  t  1
f t  2
t 1
What percent of the normal level of oxygen
exists in the pond after 1 week? After 2
weeks? After 10 weeks? What is the limit as
t approaches infinity?
Aim: Limits at Infinity
Course: Calculus
Model Problem 1
after 1 week?
1   1  1 1

f  1 

2
2
 1  1
2
50%
After 2 weeks?
2   2  1 3

f  2 

2
5
 2  1
2
60%
After 10 weeks?
10    10   1 91

f  10  

2
101
 10   1
2
90.1%
What is the limit as t approaches infinity?
Aim: Limits at Infinity
Course: Calculus
Model Problem 1
What is the limit as t approaches infinity?
t  t 1
lim f  t   lim 2
x 
x  t  1
2
t2 t 1
 2 2
2
t
lim t 2 t
x 
t
1
 2
2
t
t
divide all terms by highest
power of t in denominator
1 1
1  2
t t
 lim
x 
1
1 2
t
c
lim r  0
x  x
1 0 0

1
1 0
Aim: Limits at Infinity
Course: Calculus
Limits at Infinity
Let L be a real number.
1. The statement lim f  x   L
x 
means that for each  > 0 there exists
an M > 0 such that f  x   L  
whenever x > M.
2. The statement lim f  x   L
x 
means that for each  > 0 there exists
an N < 0 such that f  x   L  
whenever x < N.
Aim: Limits at Infinity
Course: Calculus
Limits at Infinity
f(x) is within  units of L as x  
lim f  x   L
x 


L
M
For a given positive number  there
exists a positive number M such that,
for x > M, the graph of f will lie
between the horizontal lines given by
y = L +  and y = L - .
Aim: Limits at Infinity
Course: Calculus