1.2 finding limits
DESCRIPTION
1.2 Finding Limits. Numerically and Graphically. Limits. A function f(x) has a limit L as x approaches c if we can get f(x) as close to c as possible but not equal to c. x is very close to, not necessarily at, a certain number c NOTATION:. 3 Ways to find Limits. - PowerPoint PPT PresentationTRANSCRIPT
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1.2 FINDING LIMITS
Numerically and Graphically
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Limits• A function f(x) has a limit L as x approaches c if we can get
f(x) as close to c as possible but not equal to c.
x is very close to, not necessarily at, a certain number c
NOTATION:
limx c
f (x)
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3 Ways to find Limits
• Numerically - construct a table of values and move arbitrarily close to c
• Graphically - exam the behavior of graph close to the c
• Analytically
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1) Given , find
x 1.9 1.99 1.999 1.9999
f (x)
x 2.0001 2.001 2.01 2.1
f (x)
2
24
4
4
3.61 3.9601 3.996001 3.99960001
4.00040001 4.004001 4.0401 4.41
f (x) x 2
limx 2x 2
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2) Given , find 1
1)(
3
x
xxf )(lim
1xf
x
x 0.9 0.99 0.999 0.9999
f (x)
x 1.0001 1.001 1.01 1.1
f (x)
1
13
3
3
2.710 2.9701 2.997001 2.99970001
3.00030001 3.003001 3.0301 3.31
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3. What does the following table suggest about
a)
b)
)(1
limxf
x
)(1
limxf
x
x 0.9 0.99 0.999 1.001 1.01 1.1
F(x) 7 25 4317 3.0001 3.0047 3.01
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Finding Limits Graphically• There is a hole in the graph.
Limits that Exist even though the function fails to Exist
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One sided Limits
notation
1.Limits from the right
1.Limits from the left
)(lim
xfcx
)(lim
xfcx
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4) Use the graph of to find
3
f ( x ) x 2 2
limx 1
( x 2 2)
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5) Use the graph of to find
21
1)(
2
x
xxf
1
1lim
2
1
x
xx
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0 1
0 1)(
x
xxf
)(lim0
xfx
limx 0
f (x)
6) Use the graph of to find
1
–1
1–1
Does Not Exist – DNE
limx 0
f (x)
limx 0
f (x)
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Limits that Fail to Exist
• In order for a limit to exist the limit must be the same from both the left and right sides.
1
–1
1–1
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Limits that Fail to Exist
• The behavior is unbounded or approaches an asymptote
1
–1
1–1
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Limits that Fail to Exist
• The behavior oscillates
xx
1sin
0
lim
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HOMEWORK
Page 54
# 1-10 all numerically
# 11 – 26 all graphically