Limits at Infinity

1. Limits at Infinity Lesson 4.5

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

3. Limits with Infinity • What happens to a function in the long run N1

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

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

6. Rational Functions • Leading terms dominate • m = n => limit = an/bm • m > n => limit = 0 • m < n => asymptote linear diagonal or higher power polynomial

7. Rational Functions • Vertical asymptotes • where denominator = 0 • Y-intercepts • where x = 0 • X-intercepts • where numerator = 0

8. Example • Find • horizontal asymptote • vertical asymptote(s) • zeros • y-intercept

9. Example • Find • horizontal asymptote • vertical asymptote(s) • zeros • y-intercept

10. 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

11. Assignment • Lesson 4.5 • Page 245 • Exercises 1 – 57 EOOAlso 99, 102