Evaluating Limits Analytically Lesson 1.3
Dec 26, 2015
2
What Is the Squeeze Theorem?
Today we look at various properties of limits, including the Squeeze Theorem
Today we look at various properties of limits, including the Squeeze Theorem
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How do we evaluate limits?
• Numerically– Construct a table of values.
• Graphically– Draw a graph by hand or use TI’s.
• Analytically– Use algebra or calculus.
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Properties of Limits The Fundamentals
bbcx
lim cxcx
lim nn
cxcx
lim
Basic Limits:
Let b and c be real numbers and
let n be a positive integer:
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Properties of Limits Algebraic Properties
Algebraic Properties of Limits:
Let b and c be real numbers, let n be a positive integer, and let f and g be functions
with the following properties:
Too many to fit on this page….
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Properties of Limits Algebraic Properties
Lxfcx
)(lim Kxgcx
)(limLet: and
bLxbfcx
)(lim
KLxgxfcx
)()(lim
LKxgxfcx
)()(lim
Scalar Multiple:
Sum or Difference:
Product:
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Properties of Limits Algebraic Properties
Lxfcx
)(lim Kxgcx
)(limLet: and
Quotient:
Power:
0;)(
)(lim
KK
L
xg
xfcx
nn
cxLxf
)(lim
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Evaluate by using the properties of limits. Show each step and
which property was used.
34lim 2
2x
x
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Examples of Direct Substitution - EASY
3
5
9
2
2
2
2
lim
lim3
lim 4 3
1lim
1
x
x
x
x
x
x
x x
x
33
2
2
1 1(5)
5 125
3
4(2) 3 19
2 2 15
2 1
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Properties of Limits nth roots
Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid
for all c > 0 if n is even…
nn
cxcx
lim
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Properties of Limits Composite Functions
If f and g are functions such that…
Lxgcx
)(lim )()(lim LfxfLx
and
then…
)()(lim)(lim Lfxgfxgfcxcx
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Example:
3 2
3102lim x
x
By now you should have already arrived at the conclusion that many algebraic functions can be evaluated by direct substitution.
The six basic trig functions also exhibit this desirable characteristic…
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Properties of Limits Six Basic Trig Function
Let c be a real number in the domain of the
given trig function.
cxcx
sinsinlim
cxcx
coscoslim
cxcx
tantanlim
cxcx
secseclim
cxcx
csccsclim
cxcx
cotcotlim
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A Strategy For Finding Limits
• Learn to recognize which limits can be evaluated by direct substitution.
• If the limit of f(x) as x approaches c cannot be evaluated by direct substitution, try to find a function g that agrees with f for all x other than x = c.
• Use a graph or table to find, check or reinforce your answer.
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Some Examples
• Consider
– Why is this difficult?
• Strategy: simplify the algebraic fraction
2
2
6lim
2x
x x
x
2
2 2
2 36lim lim
2 2x x
x xx x
x x
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Reinforce Your Conclusion
• Graph the Function– Trace value close to
specified point
• Use a table to evaluateclose to the point inquestion
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Find each limit, if it exists.3
1
11. lim
1x
x
x
2
1
1 1lim
1x
x x x
x
2
1lim 1x
x x
3
Direct Substitution doesn’t work!
Factor, cancel, and try again!
D.S.
Don’t forget, limits can never be undefined!
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Find each limit, if it exists.
0
1 12. lim
x
x
x
1 1x
1 1x
Direct Substitution doesn’t work.
Rationalize the numerator.
0
1 1lim
1 1x
x
x x
0lim
1 1x
x
x x
0
1lim
1 1x x
1
2
D.S.
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Special Trig Limits
0
tan43. lim
x
x
x
0
sinlimx
Ax
Ax
0
1 coslimx
Ax
Ax
1 0
0
cos 1limx
Ax
Ax
0
0
sin4 1lim
cos4x
x
x x
0
sin4 1lim
cos4x
x
x x
0
sin4 4lim
4 cos4x
x
x x
4
4
0 0
sin4 4lim lim
4 cos4x x
x
x x
1 4 4Trig limit
D.S.
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Evaluate:
2 2
0 0
2 2
0 0 0
2 3 18 2(9 6 ) 18lim lim
18 12 2 18 12 2 2 (6 )lim lim lim
h h
h h h
h h h
h h
h h h h h h
h h h
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Evaluate:
2 2
0 0
2 2
0 0 0
0
2 3 18 2(9 6 ) 18lim lim
18 12 2 18 12 2 2 (6 )lim lim lim
lim 2(6 )
h h
h h h
h
h h h
h h
h h h h h h
h h hh
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Evaluate:
2 2
0 0
2 2
0 0 0
0
2 3 18 2(9 6 ) 18lim lim
18 12 2 18 12 2 2 (6 )lim lim lim
lim 2(6 ) 12
h h
h h h
h
h h h
h h
h h h h h h
h h hh
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• Note possibilities for piecewise defined functions. Does the limit exist?
2
2
3 2 2( )
5 2
lim ( ) ?x
x if xf x
x if x
f x
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Three Special Limits
• Try it out!
0
sin 4lim ?
9x
x
x 20
1 coslimx
x
x
1
0 0 0
sin 1 coslim 1 lim 0 lim 1 xx x x
x xx e
x x
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x
xx 7sin
9sinlim
0
xxxx
x
xx
7799
7sin
9sinlim
0
xx
x
xx
x
x
77sin
7
99sin
9lim
0
xxxx
x
x
77sin
lim
99sin
lim
7
9
0
0
7
9
1
1
7
9
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Squeeze Rule
• Given g(x) ≤ f(x) ≤ h(x) on an open interval containing cAnd …
– Then
lim ( ) lim ( )
lim ( )
x c x c
x c
g x h x L
f x L
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Common Types of Behavior Associated with the Nonexistence of
a Limit1. f(x) approaches a different number from
the right side of c than it approaches from the left side.
2. f(x) increases or decreases without bound as x approaches c.
3. f(x) oscillates between 2 fixed values as x approaches c.