17 thru 25 (odd problems only)

2.2 Definitions of Limits 69 17. Finding limits from a graph Use the graph of f in the figure to find the following values or state that they do not exist. If a limit 23. S(x) = x - 25 x - 5 :0-5 does not exist, explain why. a. f(1) b. lim f(x) c. lim f(x) T 24. f(x) x - 100 Vx - 10 4 = 100 d. lim f(x) e. f(3) f. lim f(x) g. lim, f(x) h. lim f(x) *+x - 2 i. f(2) 25. f(x) = x - 1 j. lim f(x) k. lim f(x) 1. lim f(x) 26. f(x) = - 1 -14 YA x2 - 1:0 a=1 y = f ( x ) tech T 27-32. Estimating limits graphically and numerically Use a graph of f to estimate lim f(x) or to show that the limit does not exist, Eval- uate f(x) near x = a to support your conjecture, 27. f ( x ) = Inx - 29= 2 28. f ( x ) = ex - 2x - 1 x 2 Cia = 0 29. f (x ) = 1 - cos(2x - 2) (x - 1)2 ; a = 1 18. One-sided and two-sided limits Use the graph of g in the figure 30. f(x ) = 3 sin x - 2 cos x + 2 to find the following values or state that they do not exist. If a ; a =0 limit does not exist, explain why. a. 8(2) b. lim g(x) 31. f(x) = sin (x + 1 ) c. lim, g(x) ;a = -1 x-2+ d. lim g(x) e. g (3) lim g(x) x 23 x3 - 4x2 + 3x g. lim g(x) h. 8 (4) i. lim 8(x) 32 . f ( x ) = : |x - 31 ; a = 3 YA 33. Explain why or why not Determine whether the following state- ments are true and give an explanation or counterexample. a. The value of lim - x2 - 9 13 x - 3 does not exist. y = 8(x) b. The value of lim f(x) is always found by computing f(a). c. The value of lim f(x) does not exist if f(a) is undefined. d. lim Vx = 0. (Hint: Graph y = Vx.) * - 0 e. lim cot x = 0. (Hint: Graph y = cot x.) 34. The Heaviside function The Heaviside function is used in engi- neering applications to model flipping a switch. It is defined as Practice Exercises H (x ) = so ifx -1' ;a =-1 the United States is $0.47 for the first ounce (up to and including 1 oz) plus $0.21 for each additional ounce (up to and including 20. f(x ) = 3 -* ifx 2' ; a = 2 a. Graph the function p = f(w) that gives the postage p for sending a letter that weighs w ounces, for 0 4 c. Does lim f(w) exist? Explain. 22. f (x ) = /x + 21 + 2;a=-2