Evaluating Limits Analytically

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Evaluating Limits Analytically

• Lesson 1.3

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What Is 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 TIs.
• Analytically
• Use algebra or calculus.

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Properties of Limits The Fundamentals
Basic Limits Let b and c be real numbers and
let n be a positive integer
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Examples
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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
Let
and
Scalar Multiple
Sum or Difference

Product
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Properties of Limits Algebraic Properties
Let
and


Quotient


Power

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Evaluate by using the properties of limits. Show
each step and which property was used.
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Examples of Direct Substitution - EASY
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Examples
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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 gt 0 if n is even
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Properties of Limits Composite Functions
If f and g are functions such that
and
then
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Example
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.
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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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The Squeeze Theorem
FACT If
for all x on
and
then,
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Example
GI-NORMOUS PROBLEMS!!!
Use Squeeze Theorem!
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Example

• Use the squeeze theorem to find

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Properties of Limits Two Special Trig Function
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General Strategies
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Some Examples

• Consider
• Why is this difficult?
• Strategy simplify the algebraic fraction

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Reinforce Your Conclusion

• Graph the Function
• Trace value close tospecified point
• Use a table to evaluateclose to the point
  inquestion

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Find each limit, if it exists.
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Find each limit, if it exists.
Dont forget, limits can never be undefined!
Direct Substitution doesnt work!
Factor, cancel, and try again!
D.S.
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Find each limit, if it exists.
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Find each limit, if it exists.
Direct Substitution doesnt work.
Rationalize the numerator.
D.S.
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• Special Trig Limits

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• Special Trig Limits

Trig limit
D.S.
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Evaluate in any way you chose.
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Evaluate in any way you chose.
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Evaluate in any way you chose.
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Evaluate in any way you chose.
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Evaluate by using a graph. Is there a better
way?
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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Evaluate
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• Note possibilities for piecewise defined
  functions. Does the limit exist?

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Three Special Limits

• Try it out!

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Squeeze Rule

• Given g(x) f(x) h(x) on an open interval
  containing cAnd
• Then

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Common Types of Behavior Associated with the
Nonexistence of a Limit

• f(x) approaches a different number from the right
  side of c than it approaches from the left side.
• f(x) increases or decreases without bound as x
  approaches c.
• f(x) oscillates between 2 fixed values as x
  approaches c.

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• Gap in graph Asymptote
• Oscillates

c
c
c