Limits at Infinity Lesson 4.5

What Happens? We wish to investigate what happens when functions go … To infinity and beyond …

Limits with Infinity What happens to a function in the long run N1N1

Rules for Manipulating Limits Note rules on page 239 Note special limits r is a positive rational number

Manipulating, Evaluating Symbolically Use Calculator limit((x+2)/(3x-5),x,+  ) Graph and observe go to zero

Rational Functions Leading terms dominate  m = n => limit = a n /b m  m > n => limit = 0  m asymptote linear diagonal or higher power polynomial

Rational Functions Vertical asymptotes  where denominator = 0 Y-intercepts  where x = 0 X-intercepts  where numerator = 0

Example Find  horizontal asymptote  vertical asymptote(s)  zeros  y-intercept

Example Find  horizontal asymptote  vertical asymptote(s)  zeros  y-intercept

Limits Involving Trig Functions Consider f(x) = sin x  As x gets very large, function oscillates between 1 and -1  Thus no limit Consider  Squeeze theorem applies  Limit is 0

Assignment Lesson 4.5 Page 245 Exercises 1 – 57 EOO Also 99, 102