Evaluating limits analytically l.jpg
This presentation is the property of its rightful owner.
Sponsored Links
1 / 62

Evaluating Limits Analytically PowerPoint PPT Presentation


  • 222 Views
  • Updated On :
  • Presentation posted in: General

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

Download Presentation

Evaluating Limits Analytically

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Evaluating limits analytically l.jpg

Evaluating Limits Analytically

Lesson 1.3


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.


Properties of limits the fundamentals l.jpg

Properties of LimitsThe Fundamentals

Basic Limits:

Let b and c be real numbers and

let n be a positive integer:


Examples l.jpg

Examples:


Properties of limits algebraic properties l.jpg

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….


Properties of limits algebraic properties7 l.jpg

Properties of LimitsAlgebraic Properties

Let:

and

Scalar Multiple:

Sum or Difference:

Product:


Properties of limits algebraic properties8 l.jpg

Properties of LimitsAlgebraic Properties

Let:

and

Quotient:

Power:


Evaluate by using the properties of limits show each step and which property was used l.jpg

Evaluate by using the properties of limits. Show each step and which property was used.


Examples of direct substitution easy l.jpg

Examples of Direct Substitution - EASY


Examples11 l.jpg

Examples


Properties of limits n th roots l.jpg

Properties of Limitsnth 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…


Properties of limits composite functions l.jpg

Properties of LimitsComposite Functions

If f and g are functions such that…

and

then…


Example l.jpg

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…


Properties of limits six basic trig function l.jpg

Properties of LimitsSix Basic Trig Function

Let c be a real number in the domain of the

given trig function.


A strategy for finding limits l.jpg

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 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.


The squeeze theorem l.jpg

The Squeeze Theorem

FACT:

If

for all x on

and

then,


Example18 l.jpg

Example:

GI-NORMOUS PROBLEMS!!!

Use Squeeze Theorem!


Example21 l.jpg

Example:

Use the squeeze theorem to find:


Properties of limits two special trig function l.jpg

Properties of LimitsTwo Special Trig Function


General strategies l.jpg

General Strategies


Some examples l.jpg

Some Examples

  • Consider

    • Why is this difficult?

  • Strategy: simplify the algebraic fraction


Reinforce your conclusion l.jpg

Reinforce Your Conclusion

  • Graph the Function

    • Trace value close tospecified point

  • Use a table to evaluateclose to the point inquestion


Find each limit if it exists l.jpg

Find each limit, if it exists.


Find each limit if it exists27 l.jpg

Don’t forget, limits can never be undefined!

Find each limit, if it exists.

Direct Substitution doesn’t work!

Factor, cancel, and try again!

D.S.


Find each limit if it exists28 l.jpg

Find each limit, if it exists.


Find each limit if it exists29 l.jpg

Find each limit, if it exists.

Direct Substitution doesn’t work.

Rationalize the numerator.

D.S.


Slide30 l.jpg

Special Trig Limits


Slide31 l.jpg

Special Trig Limits

Trig limit

D.S.


Evaluate in any way you chose l.jpg

Evaluate in any way you chose.


Evaluate in any way you chose33 l.jpg

Evaluate in any way you chose.


Evaluate in any way you chose34 l.jpg

Evaluate in any way you chose.


Evaluate in any way you chose35 l.jpg

Evaluate in any way you chose.


Evaluate by using a graph is there a better way l.jpg

Evaluate by using a graph. Is there a better way?


Evaluate l.jpg

Evaluate:


Evaluate43 l.jpg

Evaluate:


Evaluate44 l.jpg

Evaluate:


Evaluate45 l.jpg

Evaluate:


Evaluate46 l.jpg

Evaluate:


Evaluate47 l.jpg

Evaluate:


Evaluate48 l.jpg

Evaluate:


Evaluate49 l.jpg

Evaluate:


Evaluate50 l.jpg

Evaluate:


Evaluate51 l.jpg

Evaluate:


Evaluate52 l.jpg

Evaluate:


Evaluate53 l.jpg

Evaluate:


Evaluate54 l.jpg

Evaluate:


Evaluate55 l.jpg

Evaluate:


Evaluate56 l.jpg

Evaluate:


Slide57 l.jpg

  • Note possibilities for piecewise defined functions. Does the limit exist?


Three special limits l.jpg

Three Special Limits

  • Try it out!


Squeeze rule l.jpg

Squeeze Rule

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

    • Then


Common types of behavior associated with the nonexistence of a limit l.jpg

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

c

Gap in graphAsymptote

Oscillates

c


  • Login