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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:

Too many to fit on this page….

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

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…

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.

Example:

Use the squeeze theorem to find:

Some Examples

- Consider
- Why is this difficult?
- Strategy: simplify the algebraic fraction

Reinforce Your Conclusion

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

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.

Three Special Limits

- 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.

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