1.2 Finding Limits
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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:
limx c
f (x)
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
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
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
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
Finding Limits Graphically• There is a hole in the graph.
Limits that Exist even though the function fails to Exist
One sided Limits
notation
1.Limits from the right
1.Limits from the left
)(lim
xfcx
)(lim
xfcx
4) Use the graph of to find
3
f ( x ) x 2 2
limx 1
( x 2 2)
5) Use the graph of to find
21
1)(
2
x
xxf
1
1lim
2
1
x
xx
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)
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
Limits that Fail to Exist
• The behavior is unbounded or approaches an asymptote
1
–1
1–1
Limits that Fail to Exist
• The behavior oscillates
xx
1sin
0
lim
HOMEWORK
Page 54
# 1-10 all numerically
# 11 – 26 all graphically