2.2: LIMITS INVOLVING INFINITY Objectives: • Students will be able to evaluate limits as • Students will be able to find horizontal and vertical asymptotes • Students will be able to evaluate infinite limits x
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2.2: LIMITS INVOLVING INFINITY
Objectives:• Students will be able to evaluate limits as • Students will be able to find horizontal and vertical
asymptotes• Students will be able to evaluate infinite limits
x
Finite Limits as
Given f(x) = 1/x
xx
1lim
xx
1lim
x
Look at the graph and table of values for the graph of
What is ?
What else does this tell us?
1
3)(
2
2
x
xxf
)(lim xfx
Definition
The line y=b is a horizontal asymptote of the graph of a function y= f(x) if either
OR
(Note…a graph can have at most 2 HA’s)
bxfx
)(lim bxfx
)(lim
The properties of limits as x ±∞ are on p. 67(same as properties of other limits)Evaluate the limit. Identify any horizontal asymptotes.
xx
12lim
xx
12lim
Theorem
r is a positive #, c is any real #
0lim
0lim
rx
rx
x
c
x
c
Evaluate
1.
2.
1
23lim
25lim
2
x
x
x
x
x
Uh oh…we have . This is indeterminate form. What do we do???
To find finite limits in rational functions…..Divide both the numerator and the denominator by the highest power of x in the denominator. Want to get numerator and denominator in the form then evaluate limitrx
c
Evaluate the limit. Identify the HA.
13
52lim
13
52lim
13
52lim
2
3
2
2
2
x
x
x
x
x
x
x
x
x
Extra examples??
3
2
2
3
2
lim.3
2
3lim.2
1
2lim.1
x
x
x
x
x
x
x
x
Prize!!!
What is the domain of the following function? You may not use a calculator. You will be disqualified if you do.
94)( 2 xxf
Shortcuts for Finding HA and for rational functions 1. If degree of numerator is < degree of denominator,
the limit is 0
2. If the degree of numerator = degree of denominator, the limit is the ratio of leading coefficients
3. If the degree of numerator > degree of denominator, the limit DNE
)(lim xfx
Examples. Evaluate limit and identify HA.
4
23
2
5
2
2
4lim.3
35lim.2
154
23lim.1
x
xx
x
x
xx
xx
x
x
x
Functions with 2 HA’s
Identify the Horizontal Asympotes. Prove using a limit.
12
232
x
xy
2,0 xxxFor 2,0 xxxFor
Evaluate
a.)
b.)
a.)
b.)
1
2lim
1
2lim
2
2
x
x
x
x
x
x
2
13lim
2
13lim
x
x
x
x
x
x
Use a graph or table to evaluate. 1.
2.
3.
x
x
x
x
x
x
x
sinlim
coslim
sinlim
Examples.
1.
2.
3.
xx
x
x
x
xx
x
x
x
22
sinlim
)1
cos(lim
sin5lim
Infinite Limits as x a
If the values of a function outgrow all positive bounds as x approaches a finite number a, then
If the values of a function outgrow all negative bounds as x approaches a finite number a, then
)(lim xfax
)(lim xfax
Vertical Asymptote
The line x = a is a vertical asymptote of the graph of a function y=f(x) if either
OR
)(lim xfax
)(lim xfax
Find the vertical asymptotes (if any) of the graph of the function. Prove using a limit. 1.
2.
3.
4.
)1(
2)(
16
4)(
25
158)(
1)(
2
2
2
2
2
xx
xxf
x
xxf
x
xxxf
xxf
Find the limit!!!!!(pick #’s very close to a)
1.
2.
3.
x
xx
xx
x
x
x
x
x
cos
2lim
6
32lim
2
3lim
2
2
2
3
2
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