sect.1.5 infinite limits
DESCRIPTION
Sect.1.5 Infinite Limits. Limits at infinity Infinite limits. is undefined at x = 3. From the graph. Consider the rational function. And Table. Left:. Decreases without bound. Right:. Increases without bound. Now find the Limit graphically. NOTE: the function increases - PowerPoint PPT PresentationTRANSCRIPT
Sect.1.5 Infinite Limits
Limits at infinity
Infinite limits y
)(lim xfx
Consider the rational function 3
12)(
x
xxf
)(xf
)(xf is undefined at x = 3
From the graph And Table
x 2.9 2.99 2.999 3
-68 -698 -6998 ?)(xf
3.0001 3.001 3.01 3.1 x
? 7002 702 72 )(xf
Left:
Right:
Decreases without bound
Increases without bound
)(xf
• Now find the Limit graphically
3
12lim
3 x
xx
3
12lim
3 x
xx
3
12lim
3 x
xx
NOTE: the function increasesor decreases without bound
NOTE: not the same infinite limits do not exist
LimitsLimits at infinity
Infinite Limits
lim
x nn
0 andlim
x nn
0
nx
1lim
Identifying Infinite Limits
Rational functions with undefined points in the denominator
• Numerically: table of values
• Graphically: examine end behavior
• Analytically: determine 0or 0
1) Find 2
3
2
3
9lim
lim
x
x
x
x
2
2
3 9lim
x
xx
Is the denominator approaching 0- or 0+
2
2
39
3
0
9
Examining the behavior of the denominator
Test Point0 3
2) Find
x
x
x
x
1lim
2lim
1
1
x
xx
1
2lim
1
11
12
0
3
Is the denominator approaching 0- or 0+
Test Point
Infinite Properties
• Let c and L be real numbers and let f and g be functions such that Lxgxf
cx
)(lim and )(lim
cx
limx c
g(x)
f (x)L
0
)]()([lim xgxfcx
limx c
[ f (x) g(x)]
)]()([lim xgxfcx
L < 0
L > 0
3) Find if and
76)( 2 xxxg2
1)(
x
xf)()(lim2
xgxfx
2
76lim
2
2 x
xxx
2)2(
7)2(6)2( 2
0
9
denominator
4) Find
)32)(12(
)13)(12(lim
2
1
xx
xx
x
344
16lim
2
2
2
1
xx
xx
x
3
21
421
4
121
21
6
2
2
5) Find
xx
x
1lim 2
0
HOMEWORK
Page 88
# 37-43, 45, 47, 48, and 70 all analytically