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Warmup – No calculator
bx
axsin lim )3
0
x
7x
sin lim )1
0
xx
x
sin lim )2
2
0
xx
4) Find the average speed in ft/sec of a ball modeled by over the time period [2,6] (feet1062 xxy
1f x
x
Make a table using your calculator
x f(x)
1
10
100
1000
Lets push the value ofx towards infinite
1lim 0x x
1f x
x
1lim 0x x
As the denominator gets larger, the value of the fraction gets smaller.
There is a horizontal asymptote if:
limx
f x b
or limx
f x b
2lim
1x
x
x
Example 1:
2limx
x
x
This number becomes insignificant as .x
limx
x
x 1
There is a horizontal asymptote at 1.
sin xf x
x
Example 2:
sinlimx
x
x Find:
When we graph this function, the limit appears to be zero.1 sin 1x
so for :0x 1 sin 1x
x x x
1 sin 1lim lim limx x x
x
x x x
sin0 lim 0
x
x
x
by the sandwich theorem:
sinlim 0x
x
x
Infinite Limits (Vertical Asymptotes):
1f x
x
0
1limx x
As the denominator approaches zero, the value of the fraction gets very large.
If the denominator is positive then the fraction is positive.
0
1limx x
If the denominator is negative then the fraction is negative.
vertical asymptote at x=0.
145
7)()1
2
xx
xxf
Determine Vertical Asymptotes and evaluate each limit
)( d) )()
)(c) )()
limlim
limlim
22
77
xfxfb
xfxfa
xx
xx
Example 4:
20
1limx x
20
1limx x
The denominator is positive in both cases, so the limit is the same.
20
1 limx x
Example:
Sketch a function f(x) that has all of the following properties:
3)(
3)(
)(
)(
4)2(
lim
lim
lim
lim
4
4
xf
xf
xf
xf
f
x
x
x
x