Transcript Chapter 2.2
AP CALCULUS AB
Chapter 2:
Limits and Continuity
Section 2.2:
Limits Involving Infinity
What you’ll learn about
Finite Limits as x→±∞
Sandwich Theorem Revisited
Infinite Limits as x→a
End Behavior Models
Seeing Limits as x→±∞
…and why
Limits can be used to describe the behavior
of functions for numbers large in absolute
value.
Finite limits as x→±∞
The symbol for infinity (∞) does not represent a real number.
We use ∞ to describe the behavior of a function when the
values in its domain or range outgrow all finite bounds.
For example, when we say “the limit of f as x approaches
infinity” we mean the limit of f as x moves increasingly far to
the right on the number line.
When we say “the limit of f as x approaches negative
infinity (- ∞)” we mean the limit of f as x moves increasingly
far to the left on the number line.
Horizontal Asymptote
The line y b is a horizontal asymptote of the graph of a function
y f x if either
lim f x b
x
or
lim f x b
x
Example Horizontal Asymptote
Use a graph and tables to find a lim f x and
x
c Identify all horizontal asymptotes.
f x
f x .
b xlim
x 1
x
f x 1
a lim
x
f x 1
b xlim
c Identify all horizontal asymptotes.
[-6,6] by [-5,5]
y 1
Section 2.2 – Limits Involving Infinity
To find Horizontal
Asymptotes:
Divide numerator
and denominator by
the highest power of
x.
Note:
1
lim 0
x x
x 3 3x 2 5
Ex : lim
x 2 x 3 x 1
x 3 3x 2 5
3 3
3
x
lim x 3 x
x 2 x
x 1
3 3
3
x
x
x
3 5
1 3
x x
lim
x
1
1
2 2 3
x
x
1 0 0 1
200 2
Example Sandwich Theorem
Revisited
The sandwich theorem also holds for limits as x .
cos x
Find lim
graphically and using a table of values.
x
x
The graph and table suggest that the function oscillates about the x -axis.
cos x
Thus y 0 is the horizontal asymptote and lim
0
x
x
Properties of Limits as x→±∞
If L, M and k are real numbers and
lim f x L
x
1.
Sum Rule :
and
lim g x M , then
x
lim f x g x L M
x
The limit of the sum of two functions is the sum of their limits.
2.
Difference Rule :
lim f x g x L M
x
The limit of the difference of two functions is the difference
of their limits
Properties of Limits as x→±∞
3.
Product Rule:
lim f x g x L M
x
The limit of the product of two functions is the product of their limits.
4.
Constant Multiple Rule:
lim k f x k L
x
The limit of a constant times a function is the constant times the limit
of the function.
5.
Quotient Rule :
lim
x
f x
g x
L
, M 0
M
The limit of the quotient of two functions is the quotient
of their limits, provided the limit of the denominator is not zero.
Properties of Limits as x→±∞
6.
If r and s are integers, s 0, then
Power Rule :
r
s
r
s
lim f x L
x
r
s
provided that L is a real number.
The limit of a rational power of a function is that power of the
limit of the function, provided the latter is a real number.
Infinite Limits as x→a
If the values of a function f ( x) outgrow all positive bounds as x approaches
a finite number a, we say that lim f x . If the values of f become large
xa
and negative, exceeding all negative bounds as x approaches a finite number a,
we say that lim f x .
xa
Vertical Asymptote
The line x a is a vertical asymptote of the graph of a function
y f x if either
lim f x or lim f x
x a
x a
Example Vertical Asymptote
Find the vertical asymptotes of the graph of f ( x) and describe the behavior
of f ( x) to the right and left of each vertical asymptote.
8
f x
4 x2
The values of the function approach to the left of x 2.
The values of the function approach + to the right of x 2.
The values of the function approach + to the left of x 2.
The values of the function approach to the right of x 2.
8
8
lim
and
lim
2
2
x 2 4 x
x 2 4 x
8
8
lim
and
lim
2
2
x2 4 x
x 2 4 x
So, the vertical asymptotes are x 2 and x 2
[-6,6] by [-6,6]
Section 2.2 – Limits Involving Infinity
1.
2.
To find vertical asymptotes:
Cancel any common factors in the numerator and
the denominator
Set the denominator equal to 0 and solve for x.
x2 4
x 2x 2
Ex : 2
x x 2 x 1x 2
The vertical asymptote is x=-1. (from denominator)
There is a hole at x=2. (from the cancelled factor)
The x-intercept is at x=-2. (from numerator)
End Behavior Models
The function g is
a a right end behavior model for f if and only if lim
x
b a left end behavior model for f if and only if lim
x
f x
g x
f x
g x
1.
1.
Example End Behavior Models
Find an end behavior model for
3x 2 2 x 5
f x
4 x2 7
Notice that 3 x 2 is an end behavior model for the numerator of f , and
4 x 2 is one for the denominator. This makes
3x 2 3
= an end behavior model for f .
2
4x 4
End Behavior Models
If one function provides both a left and right end behavior model, it is simply called
an end behavior model.
In general, g x an x n is an end behavior model for the polynomial function
f x an x n an 1 x n 1 ... a0 , an 0
Overall, all polynomials behave like monomials.
End Behavior Models
3
is also a horizontal
4
asymptote of the graph of f . We can use the end behavior model of a
In this example, the end behavior model for f , y
rational function to identify any horizontal asymptote.
A rational function always has a simple power function as
an end behavior model.
Example “Seeing” Limits as
x→±∞
We can investigate the graph of y f x as x by investigating the
1
graph of y f as x 0.
x
1
Use the graph of y f to find lim f x and lim f x
x
x
x
1
for f x x cos .
x
cos x
1
The graph of y f =
is shown.
x
x
1
lim f x lim f
x
x 0
x
1
lim f x lim f
x
x 0
x
Section 2.2 – Limits Involving Infinity
Definition of Infinite Limits:
A limit in which f(x) increases or
decreases without bound as x approaches
c is called an infinite limit.
Section 2.2 – Limits Involving Infinity
lim f ( x)
x c
lim f ( x)
x c
c
So lim f ( x) DNE
x c
(Does not Exit)
Section 2.2 – Limits Involving Infinity
lim f ( x)
x c
c
lim f ( x)
x c
So lim f ( x)
x c
Section 2.2 – Limits Involving Infinity
If
1.
Properties of Infinite Limits
lim f ( x) and lim g ( x) L
x c
x c
Sum or difference:
lim f ( x) g ( x) L
x c
2.
Product: lim f ( x) g ( x) if L 0
x c
lim f ( x) g ( x) if L 0
x c
3.
Quotient:
g ( x)
lim
0
x c f ( x )