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 Limits The Fundamentals Basic Limits: Let b and c be real numbers and let n be a positive integer:

Examples:

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….

Properties of Limits Algebraic Properties Let: and Scalar Multiple: Sum or Difference: Product:

Properties of Limits Algebraic Properties Let: and Quotient: Power:

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

Examples of Direct Substitution - EASY

Examples

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 > 0 if n is even…

Properties of Limits Composite 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 Limits Six 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 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.

The Squeeze Theorem FACT: If for all x on and then,

Example: GI-NORMOUS PROBLEMS!!! Use Squeeze Theorem!

Example: Use the squeeze theorem to find:

Properties of Limits Two Special Trig Function

General Strategies

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

Reinforce Your Conclusion Graph the Function Trace value close to specified point Use a table to evaluate close to the point in question

Find each limit, if it exists.

Find each limit, if it exists. Don’t forget, limits can never be undefined! Direct Substitution doesn’t work! Factor, cancel, and try again! D.S.

Find each limit, if it exists.

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 in any way you chose.

Evaluate in any way you chose.

Evaluate in any way you chose.

Evaluate in any way you chose.

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

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

Evaluate:

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

Three Special Limits Try it out!

Squeeze Rule Given g(x) ≤ f(x) ≤ h(x) on an open interval containing c And … 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.

Gap in graph Asymptote Oscillates c c c