2.2: LIMITS INVOLVING INFINITY Objectives: Students will be able to evaluate limits as Students will be able to find horizontal and vertical asymptotes.

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Presentation transcript:

2.2: LIMITS INVOLVING INFINITY Objectives: Students will be able to evaluate limits as Students will be able to find horizontal and vertical asymptotes Students will be able to evaluate infinite limits

Finite Limits as Given f(x) = 1/x

Look at the graph and table of values for the graph of What is ? What else does this tell us?

Definition The line y=b is a horizontal asymptote of the graph of a function y= f(x) if either OR (Note…a graph can have at most 2 HA’s)

The properties of limits as x ±∞ are on p. 67 (same as properties of other limits) Evaluate the limit. Identify any horizontal asymptotes.

Theorem r is a positive #, c is any real #

Evaluate Uh oh…we have. This is indeterminate form. What do we do???

To find finite limits in rational functions….. Divide both the numerator and the denominator by the highest power of x in the denominator. Want to get numerator and denominator in the form then evaluate limit

Evaluate the limit. Identify the HA.

Extra examples??

Prize!!! What is the domain of the following function? You may not use a calculator. You will be disqualified if you do.

Shortcuts for Finding HA and for rational functions 1. If degree of numerator is < degree of denominator, the limit is 0 2. If the degree of numerator = degree of denominator, the limit is the ratio of leading coefficients 3. If the degree of numerator > degree of denominator, the limit DNE

Examples. Evaluate limit and identify HA.

Functions with 2 HA’s Identify the Horizontal Asympotes. Prove using a limit.

Evaluate a.) b.) a.) b.)

Use a graph or table to evaluate

Examples

Infinite Limits as x a If the values of a function outgrow all positive bounds as x approaches a finite number a, then If the values of a function outgrow all negative bounds as x approaches a finite number a, then

Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function y=f(x) if either OR

Find the vertical asymptotes (if any) of the graph of the function. Prove using a limit

Find the limit!!!!! (pick #’s very close to a)