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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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Basic Properties and Rules
Constant rule
Limit of x rule
Scalar multiple rule
Sum rule(the limit of a sum is the sum of the limits)
limx ck k
limx cx c
lim ( ) lim ( )x c x c
k f x k f x
lim ( ) ( ) lim ( ) lim ( )x c x c x c
f x g x f x g x
See other properties pg. 79-81
See other properties pg. 79-81
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Limits of Functions
Limit of a polynomial P(x)• Can be demonstrated
using the basic properties and rules
Similarly, note the limit of a rational function
lim ( ) ( )x cP x P c
( )Given ( )
( )
( )lim ( )
( )x c
P xQ x
D x
P cQ x
D c
What stipulation must be made concerning
D(x)?
What stipulation must be made concerning
D(x)?
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Try It Out
Evaluate the limits• Justify steps using properties
3 2
0lim 5 4x
x x
1
0
sinlim
1x
x
x
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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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Some Examples
Rationalize the numerator of rational expression with radicals
Note possibilities for piecewise defined functions
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2lim
4x
x
x
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
View GraphView
GraphView
GraphView
GraphView
GraphView
Graph
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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