Sect.1.5 continued Infinite Limits Limits at infinity Infinite limits

Consider the rational function is undefined at x = 3 From the graph And Table x 2.9 2.99 2.999 3 -68 -698 -6998 ? 3.0001 3.001 3.01 3.1 x ? 7002 702 72 Left: Decreases without bound Right: Increases without bound

Now find the Limit graphically NOTE: the function increases or decreases without bound NOTE: not the same infinite limits do not exist

Limits and Notation Limits at infinity Infinite Limits

Find Is the denominator approaching 0- or 0+ Examining the behavior of the denominator Test Point

Find Is the denominator approaching 0- or 0+ Test Point

Infinite Properties Let c and L be real numbers and let f and g be functions such that L > 0 L < 0

3) Find if and Test point

4) Find

Infinite Limits and Vertical Asymptotes The line x = c is a vertical asymptote of if and

5) Find the vertical asymptote of Set denominator equal to zero Justify your answer Right Left Test point Since both limits tend to ± infinity, the line x = 2 is a Vertical Asymptote

6) Find the vertical asymptote of

6) continued Right Left Test point Since both limits tend to ±infinity, the line , where n is all integers, is a Vertical Asymptote

HOMEWORK Page 88 # 1 – 4, 13 – 23 odd, 29 – 32 all # 37-43, 45, 47, 48, and 70 Work all problems analytically