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….
Properties of LimitsAlgebraic Properties Let: and Scalar Multiple: Sum or Difference: Product:
Properties of LimitsAlgebraic Properties Let: and Quotient: Power:
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…
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.
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:
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.
Find each limit, if it exists. Direct Substitution doesn’t work. Rationalize the numerator. D.S.
Special Trig Limits Trig limit D.S.