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Can the graph of y = f (x) intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs.

(b) How many horizontal asymptotes can the graph of y = f (x) have? Sketch graphs to illustrate the possibilities.

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Sixta Kovacek
Sixta KovacekLv2
13 May 2020

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(2) Let f(x) = 578 - 6x6 - 3 4.24 + 10.03 Find all horizontal and vertical asymptotes of f. (A) Vertical asymptote at <= -, and horizontal asymptotes y = (B) Vertical asymptote at x = --, and horizontal asymptote y = 5 (C) Vertical asymptote at x = 0, -, and horizontal asymptotes y=+* (D) Vertical asymptote at x = 0, -, and horizontal asymptote y = (E) None of the above.

(1) In the space provided, sketch the graph of a function f satisfying the properties below. -00 • f is defined and differentiable for all I* 2. • f(0) = 0 . f has vertical asymptotes at I=+2 . f has horizontal asymptote y=0 as 17 00, and y=1 as 1 • f is increasing on (-2,2) . f is decreasing on (-00,-2) and (2,00) • f is concave up on (0, 2) and (2.00) • f is concave down on (-2,0) and (-00,--2)

4. (21 pts) (I) (6 pts) Determine the value of a constant b for which the function g is continuous at 0. Show your work. Justify your answer! 2 x +b if x<0, X-5 g(x) = if x 20, x#4. We have to show that lim g (x) = g (0). g(0) = 0 +18 = -1; lim X0* 0 + 16 ) = lim x + 16 **0* x2 - 160 - 16- Since lim g (x) has to exist (in order for g to be continuous at x = 0), it has to be true that x-0- x0+ X- 0 lim g(x) - lim g(x) = lim g(x). Therefore, -1, and , so b = 5. so, if b = 5, lim g (x) = g(0), and therefore, g is continuous at x = 0. (II) (4 pts) Evaluate the limit. Show your work! 2. b limx- (a) lim g(x) = lim *+- 2 x + b X - 5 : 1 * +16 10 g (x) = lim x++ - Elim x2 - 16 x++ (x - 10). OR 2x+5. 2; (a) lim g (x) = lim x--00 X- (b) lim x++ 2 + 16 g(x) = lim **** x2 - 16 X-+ (III) (4 pts) Determine whether g has any horizontal asymptotes. If so, write the equation (or equations) of all horizontal asymptotes. y = 2, by part (II) (a) y = 0, by part (II) (b) (IV) (7 pts) Determine whether g has any vertical asymptotes. If so, write the equation (or equations) of all vertical asymptotes. Show your work. Justify your answer! 5 + 16 There is no vertical asymptote at x = 5, because lim g(x) = x+5 52-16' -8 + b (x) = there is no vertical asymptote at x = -4, because lim X -4 - 4 - 5 But, lim g(x) = lim + 16 x+4+ x2 - 16 lim_ (x + 16) -20 x+4+ (x - 4) 60 (x + 4) +8 (or the numerator goes to 20, and the denominator is positive and goes to o, as x + 4*) = +00 So, the function g has only one vertical asymptote, x = 4.

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