Ms. Battaglia AB/BC Calculus

1. 1.5/3.5 Infinite LimitsObjective: Determine Infinite Limits from the left and right; determine horizontal asymptotes. Ms. Battaglia AB/BC Calculus

2. Infinite Limits Let f be the function given by 3/(x-2) A limit in which f(x) increases or decreases without bound as x approaches c is called an infinite limit. x approaches 2 from the left x approaches 2 from the right f(x) decreases without bound f(x) increases without bound

3. Definition of Infinite Limits ∞ Let f be a function that is defined at every real number in some open interval containing c (except possibly c itself). The statement means that for each M>0 there exists a δ>0 such that f(x)>M whenever 0<|x-c|<δ. Similarly, means that for each N<0 there exists a δ>0 such that f(x)<N whenever o<|x-c|<δ. To define the infinite limit from the left, replace 0<|x-c|<δ by c-δ<x<c. To define the infinite limit from the right, replace 0<|x-c|<δ by c<x<c+δ

4. Determining Infinite Limits from a Graph • Determine the limit of each function shown as x approaches 1 from the left and from the right.

5. Vertical Asymptote Definition If f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x=c is a vertical asymptote of the graph of f. Thm 1.14 Vertical Asymptotes Let f and g be continuous on an open interval containing c. If f(c)≠0, g(c)=0, and there exists on an open interval containing c such that g(x)≠0 for all x≠c in the interval, then the graph of the function given by has a vertical asymptote.

6. Finding Vertical Asymptotes Determine all vertical asymptotes of the graph of each function.

7. A Rational Function with Common Factors • Determine all vertical asymptotes of the graph.

8. Determining Infinite Limits • Find each limit.

9. Thm 1.15 Properties of Infinite Limits Let c and L be real numbers and let f and g be functions such that 1. Sum or difference: 2. Product: , L > 0 , L<0 3. Quotient: Similar properties hold for one-sided limits and for functions for which the limit of f(x) as x approaches c is -∞

10. Determining Limits Find each limit.

11. Definition of Limits at Infinity Let L be a real number. The statement means that for each ε>0 there exists an M>0 such that |f(x)-L|<ε whenever x>M. The statement means that for each ε>0 there exits an N<0 such that |f(x)-L|<ε whenever x < N.

12. Horizontal Asymptote The line y=L is a horizontal asymptote of the graph of f if or Thm3.10 Limits at Infinity If r is a positive rational number and c is any real number, then Furthermore, if xr is defined when x<0, then

13. Find the limit

14. Find the limit

15. Find each limit

16. Guidelines for Finding Limits at + of Rational Functions • If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0. • If the degree of the numerator is equal to the degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients. • If the degree of the numerator is greater thanthe degree of the denominator, then the limit of the rational function does not exist.

17. Find the limit

18. Find each limit

19. Definition of Infinite Limits at Infinity Let f be a function defined on the interval (a,∞) The statement means that for each positive number M, there is a corresponding number N>0 such that f(x)>M whenever x>N. The statement means that for each negative number M, there is a corresponding number N>0 such that f(x)<M whenever x>N. Find each limit:

20. Classwork/ Homework • Read 1.5 Page 88 #7, 9, 11, 21-49 every other odd, 65, 68, 73-76 • Read 3.5 Page 205 #1-6, 19-33 odd, 90 • Start preparing for Summer Material and Chapter 1 Test