Section 2 2 LIMITS INVOLVING INFINITY Finite Limits

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Section 2. 2 LIMITS INVOLVING INFINITY

Section 2. 2 LIMITS INVOLVING INFINITY

Finite Limits as x Approaches Infinity �Definition 1: The line y = b is

Finite Limits as x Approaches Infinity �Definition 1: The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either

Horizontal Asymptotes

Horizontal Asymptotes

Example 1 �Looking for horizontal asymptotes.

Example 1 �Looking for horizontal asymptotes.

Example 2 �Use graphs and tables to find (a) and (b). (c) Identify all

Example 2 �Use graphs and tables to find (a) and (b). (c) Identify all horizontal asymptotes.

Example 3 �Use graphs and tables to find (a) and (b). (c) Identify all

Example 3 �Use graphs and tables to find (a) and (b). (c) Identify all horizontal asymptotes.

Example 4 �Use graphs and tables to find (a) and (b). (c) Identify all

Example 4 �Use graphs and tables to find (a) and (b). (c) Identify all horizontal asymptotes.

Infinite Limits as x Approaches a �Definition 2: The line x = a is

Infinite Limits as x Approaches a �Definition 2: The line x = a is a vertical asymptote of the graph of a function y = f(x) if either

Vertical Asymptote

Vertical Asymptote

Example 5 �(a) Find the vertical asymptotes of the graph of f(x). (b) Describe

Example 5 �(a) Find the vertical asymptotes of the graph of f(x). (b) Describe the behavior of f(x) to the left and right of each vertical asymptote.

Example 6 �(a) Find the vertical asymptotes of the graph of f(x). (b) Describe

Example 6 �(a) Find the vertical asymptotes of the graph of f(x). (b) Describe the behavior of f(x) to the left and right of each vertical asymptote.

TOTD �

TOTD �

Example 7 �(a) Find the vertical asymptotes of the graph of f(x). (b) Describe

Example 7 �(a) Find the vertical asymptotes of the graph of f(x). (b) Describe the behavior of f(x) to the left and right of each vertical asymptote.

Example 8 �(a) Find the vertical asymptotes of the graph of f(x). (b) Describe

Example 8 �(a) Find the vertical asymptotes of the graph of f(x). (b) Describe the behavior of f(x) to the left and right of each vertical asymptote.

Example 9

Example 9

Example 10

Example 10

TOTD �

TOTD �

Example 11 �Show that while f and g are quite different for numerically small

Example 11 �Show that while f and g are quite different for numerically small values of x, they are virtually identical for |x| large.

End Behavior Models �The function is (a) a right-end behavior model for iff (b)

End Behavior Models �The function is (a) a right-end behavior model for iff (b) a left-end behavior model for iff

Example 12 �(a) Find a power function end behavior model for f. (b) Identify

Example 12 �(a) Find a power function end behavior model for f. (b) Identify any horizontal asymptotes.

Example 13 �(a) Find a power function end behavior model for f. (b) Identify

Example 13 �(a) Find a power function end behavior model for f. (b) Identify any horizontal asymptotes.

Example 14 �Find (a) a simple basic function as a right end behavior model

Example 14 �Find (a) a simple basic function as a right end behavior model and (b) a simple basic function as a left end behavior model for the function.

Example 15 �Find (a) a simple basic function as a right end behavior model

Example 15 �Find (a) a simple basic function as a right end behavior model and (b) a simple basic function as a left end behavior model for the function.

Example 16

Example 16

Example 17

Example 17