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Mathematics
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24 Sep 2018

1. (CONTINUED) (c) (2 pts) f(0) = - 6 (a) (2 pts) f (4) = - 6 (e) (2 pts) lim f (x) = 2 (f) (2 pts) lim f (x) = *(x)= 0 (g) (2 pts) lim f (x) = (IV) (2 pts) Is f one-to one ? Why? yes. It passes the horizontal line test (V) (4 pts) Is f continuous at a = 0? Use the definition of continuity to explain your answer.

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Jamar Ferry
Jamar FerryLv2
24 Sep 2018
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1. (20 pts) The graph of a function f is given in the figure below. Use the graph of f to answer the questions below. y = f(x) -64 - - - - - - - - - - - - (I) (1 pt) Pind the domain of f. Domain of f = (-0, 3) U (3, +co) (II) (1 pt) Find the range of f. Range of f = (-, +O) (a) (1 pt) lim f (x) = 0 (b) (1 pt) f(3) = DNE (does not exist) (c) (1 pt) f (5) = 2 (a) (1 pt) lim f (x) = 0 (0) (1 pt) lim f (x) = -- (f) (1 pt) lim f (x) = DNE 0 1 (g) (1 pt) lim f (x) = + (h) (1 pt) lim f (x) = -6 X-++00 (IV) (2 pts) Find all vertical asymptotes. (V) (2 pts) Find all horizontal asymptotes. (VI) (6 pts) Determine the intervals of continuity for f. (-0, 0], (0, 3), (3, 5), (5, +00)

4. (21 pts) Let g be the function given by ( 6 - X 2 + x if x < 2, x + -2 g(x) = x2 – x-1 if x>2. (a) (6 pts) Use the definition of continuity to determine whether the function g is continuous at 2. Show your work. (b) (3 pts) Evaluate the limit. Justify your answer. Show your work. Write "does not exist" only if the limit does not exist and is not + or -0. lim g(x) = lim -2+ = +00 X 2 + X since the form of the limit is NUM: POS, DENOM: POS (c) (2 pts) Write the equation(s) of all vertical asymptotes of g. Write "none" if appropriate. The result in part (b) implies that the line x= -2 is a vertical asymptote. This is the only vertical asymptote, since the function g is continuous on its domain (d) (2 pts) Determine the largest) intervals of continuity of g. Use interval notation to write your answer. Let g be the function given by 12+ if x < 2, x# -2 2 + x g(x)= x2-x-1 x-1 if x>2. (e) (6 pts) Evaluate each limit. Do not use L'Hôpital's Rule. Write "does not exist" only if the limit does not exist and is not too or -00. Show your work. 6-1 i lim g(x) = lim -00 2 + X = lim -002 + x 0-1 + 1 x -00 x x x -+- 2 +1 0 0 + 11 x2 - x-1 1 ii. lim g(x) = lim x++ V x2 – x-1 X-1 x2 - x-1 X - 1 = lim x +oc x +0 - lim Vam M93001 (f) (2 pts) Write the equation(s) of all horizontal asymptotes of g. Write "none" if appropriate.

4. (16 pts) MULTIPLE CHOICE! CIRCLE THE CORRECT ANSWER IN EACH PART. (I) ( 4 pts) Complete the statement of the Intermediate Value Theorem: Suppose f is continuous on the interval [a, b] and L is a number between f (a) and f (b). Then there is at least one number c in (a, b) such that (a) C = L; (b) f(c) = 0; (c) f(c) = L; (a) f' (c) = L; (e) C = 0; (f) f (a) < c < f (b); (g) NONE OF THE PREVIOUS ANSWERS. (II) (4 pts) The figure shows the graph of a function f. At what point (points) c does the conclusion of the Intermediate Value Theorem hold for f (x) on the interval [1, 5] and L = 2. --- - - - -1 - - - 2 - - I- - - (a) c = 1; (b) C = 0; (c) C = 2; (a) c = 3; (e) C = 4; (f) C = 5; (g) c= -1. (III) (4 pts) Given that cos x s f (x) s et, for all x > 0 evaluate lim f (x) x+0+ (a) lim f (x) = 0; (b) lim f (x) = 1; (c) lim f (x) = e; x0+ X0+ (d) lim f(x) DOES NOT EXIST. WE DON'T HAVE ENOUGH INFORMATION TO ANSWER THE QUESTION *40+ (IV) (4 pts) Given the function 1 f (x) = , find its inverse, f+(x) x -2 (a) f (x) = x -2; (b) f«+ (x) = x+2 ; (c) fº+ (x) = -2; (f) f-+ (x) -2; (e) f (x) DOES NOT EXIST.

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