# Evaluating Limits Analytically - PowerPoint PPT Presentation

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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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Evaluating Limits Analytically

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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

• Use algebra or calculus.

### Properties of LimitsThe Fundamentals

Basic Limits:

Let b and c be real numbers and

let n be a positive integer:

### 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:

### Properties of LimitsAlgebraic Properties

Let:

and

Scalar Multiple:

Sum or Difference:

Product:

Let:

and

Quotient:

Power:

### 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 LimitsComposite Functions

If f and g are functions such that…

and

then…

### 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 LimitsSix Basic Trig Function

Let c be a real number in the domain of the

given trig function.

### 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.

FACT:

If

for all x on

and

then,

### Example:

GI-NORMOUS PROBLEMS!!!

Use Squeeze Theorem!

### Example:

Use the squeeze theorem to find:

### Some Examples

• Consider

• Why is this difficult?

• Strategy: simplify the algebraic fraction

• Graph the Function

• Trace value close tospecified point

• Use a table to evaluateclose to the point inquestion

### Find each limit, if it exists.

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 exists.

Direct Substitution doesn’t work.

Rationalize the numerator.

D.S.

Special Trig Limits

Special Trig Limits

Trig limit

D.S.

### Evaluate:

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

• Try it out!

### 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

• 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.

c

c

Gap in graphAsymptote

Oscillates

c