2.2 Limits and Infinity
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Transcript 2.2 Limits and Infinity
Limits
Involving
Infinity
2.2
Bell work
Without a calculator, evaluate each limit either intuitively or
with a table of values.
1 . lim
x
2.
lim
1
3 . lim
x
x
x 0
1
x
4 . lim
x 0
If a limit approaches
infinity it technically
does not exist.
1
x
1
x
4
Graphically:
f
x
3
2
1
1
x
-4
-3
-2
-1
0
1
2
3
4
-1
-2
lim
x
1
0
x
-3
-4
lim
x
1
0
x
There is a horizontal asymptote at y = b if:
lim f
x
x b
or
lim f
x
x b
Asking you to find any horizontal asymptotes is the
same as asking you to find lim f x .
x
4
Graphically:
3
f
x
1
2
1
x
-4
-3
-2
-1
0
1
2
3
4
-1
lim
x 0
1
-2
x
lim
-3
x 0
-4
1
x
There is a vertical asymptote at x = a if:
lim f
x a
x
or
lim f
x a
x
More examples by graphing:
f
x
2x
x 1
lim f ( x ) 2
x
horizontal asymptote
at y = 2
lim f ( x )
x 1
vertical asymptote
at x = -1
Without graphing tell whether or not the function has a
horizontal asymptote. If yes, state the equation.
f ( x)
x
x 1
2
lim
x
x
x 1
2
lim
x
x
x
2
lim
x
x
x
1
This number becomes insignificant as x .
There is a horizontal asymptote at y = 1.
Verify graphically:
f ( x)
x
x 1
2
The other asymptote
at y = -1 can be
found similarly by
finding
lim f ( x )
x
End-Behavior
Compare the graphs.
f
x 3x
4
2 x 3x 5x 6
3
2
and
f
x
3x
4
When the value of x is
small, there is a notable
difference in y.
End-Behavior
Zoom out and compare the ends of the graphs…
f
x 3x
4
2 x 3x 5x 6
3
2
and
f
x
3x
4
When the value of x is
large, the y-values are
virtually identical.
End-Behavior
In general, g ( x ) a n x is an end behavior m odel for
n
f ( x ) a n x a n 1 x
n
n 1
... a 0 , a n 0.
This means that the end behavior model for
f
x
2 x 7 x x 3x 1
5
4
f
2
x 2x
5
is
Using end-behavior models to find horizontal asymptotes:
Given: f
x
Ax
m
Bx
n
If m n , there is N O horizontal asym ptote.
f ( x)
3x 1
5
If m n , the horizontal asym ptote is y 0.
f (x)
1
x
If m n , the horizontal asym ptote is y
3x 7 x 1
2
f ( x)
2x 3
2
A
.
B
; horizontal asym ptote y
3
2
a.) Find a power function end behavior model.
b.) Identify any horizontal asymptotes.
1. f
x 6x
2. f
x
3. f
4. f
5. f
x
x
x
2
2x 7
2 x x x 1
3
2
x3
2 x 3x 7
2
4x x 3
2
x 9
2
3x 2 x 1
3
x 1
Find the horizontal asymptotes without graphing.
f x
2x x x 1
5
4
2
3x 5x 7
2
NO HORIZONTAL ASYMPTOTE
Proof:
2x x x 1
5
lim
x
lim
x
2x
5
3x
2
4
2
3x 5x 7
2
lim
x
2
3
x
3
f
x
sin x
lim
sin x
x 0
x
1
Find:
x
lim
sin x
x
x
2
1
-12
-10
-8
-6
lim
x
-4
sin x
-2
0
-1
-2
2
4
6
8
10
appears to be 0
x
T his can be verified by the Sandw ich T he orem .
12
Find:
lim
5 x sin x
x
x
5 x sin x
lim
x
x
x
lim 5 lim
x
x
50
5
sin x
x
This also means
that there is a
horizontal
asymptote at y = 5.
Find lim
x 0
lim
x 0
lim
x 0
1
2
2
x
1
x
lim
x 0
1
x
2
The denominator is positive
in both cases, so the limit is
the same.
1
x
2
This also means
that there is a
vertical asymptote
at x = 0.
Remember that technically, this means the limit does not exist.
Often you can just “think through” limits.
0
1
lim sin
x
x
0
p
pg 71 #1-16 (calculator)
#29-38, 47-50, 23-28 (no calc)