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Evaluating Limits Analytically. Lesson 1.3. What Is the Squeeze Theorem?. Today we look at various properties of limits, including the Squeeze Theorem. How do we evaluate limits?. Numerically Construct a table of values. Graphically Draw a graph by hand or use TI’s. Analytically

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What is the squeeze theorem l.jpg
What Is the Squeeze Theorem?

Today we look at various properties of limits, including the Squeeze Theorem


How do we evaluate limits l.jpg
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 LimitsThe Fundamentals

Basic Limits:

Let b and c be real numbers and

let n be a positive integer:



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Properties of LimitsAlgebraic 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 LimitsAlgebraic Properties

Let:

and

Scalar Multiple:

Sum or Difference:

Product:


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Properties of LimitsAlgebraic Properties

Let:

and

Quotient:

Power:



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Examples of Direct Substitution - EASY and which property was used.


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Examples and which property was used.


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Properties of Limits and which property was used.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…


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Properties of Limits and which property was used.Composite Functions

If f and g are functions such that…

and

then…


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Example: and which property was used.

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 and which property was used.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 and which property was used.

  • Learn to recognize which limits can be evaluated by direct substitution.

  • If the limit of f(x) as x approaches ccannot 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 and which property was used.

FACT:

If

for all x on

and

then,


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Example: and which property was used.

GI-NORMOUS PROBLEMS!!!

Use Squeeze Theorem!


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Example: and which property was used.

Use the squeeze theorem to find:


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Properties of Limits and which property was used.Two Special Trig Function


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General Strategies and which property was used.


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Some Examples and which property was used.

  • Consider

    • Why is this difficult?

  • Strategy: simplify the algebraic fraction


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Reinforce Your Conclusion and which property was used.

  • 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. and which property was used.


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Don’t forget, limits can never be undefined! and which property was used.

Find each limit, if it exists.

Direct Substitution doesn’t work!

Factor, cancel, and try again!

D.S.


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Find each limit, if it exists. and which property was used.


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Find each limit, if it exists. and which property was used.

Direct Substitution doesn’t work.

Rationalize the numerator.

D.S.


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Special Trig Limits and which property was used.


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Special Trig Limits and which property was used.

Trig limit

D.S.


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Evaluate in any way you chose. and which property was used.


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Evaluate in any way you chose. and which property was used.


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Evaluate in any way you chose. and which property was used.


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Evaluate in any way you chose. and which property was used.


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Evaluate by using a graph. and which property was used.Is there a better way?


Evaluate l.jpg
Evaluate: and which property was used.


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Evaluate: and which property was used.


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Evaluate: and which property was used.


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Evaluate: and which property was used.


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Evaluate: and which property was used.


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Evaluate: and which property was used.


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Evaluate: and which property was used.


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Evaluate: and which property was used.


Evaluate50 l.jpg
Evaluate: and which property was used.


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Evaluate: and which property was used.


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Evaluate: and which property was used.


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Evaluate: and which property was used.


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Evaluate: and which property was used.


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Evaluate: and which property was used.


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Evaluate: and which property was used.



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Three Special Limits limit exist?

  • Try it out!


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Squeeze Rule limit exist?

  • 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.


Slide62 l.jpg

c a Limit

c

Gap in graph Asymptote

Oscillates

c


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