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Problem 1 a) (10 points! Use the limit laws and techniques for computing limits discussed in Chapter 2 to compute lim 1 . Do not use L'Hôpital's Rule or tables of values. +-+-2 (1+2) b) (12 points Use L'Hôpital's Rule to compute lim 5.+22 n(3r+11

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Lelia Lubowitz
Lelia LubowitzLv2
17 Jan 2018
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3. (18 pts) Show your work. Let f be a function defined by f(x) = x12 (a) Write the form of the limit and evaluate the limit. Do not use L'Hôpital's Rule. Show your work. 12 12 - 40=(0) - -= lim - 2 = * + 2) = Lima (68–1672 - FORM: (b) Find the natural) domain off. Write your answer in interval notation. (-0, -2) (-2, +00) (c) Let f-1 be an inverse of f. Find an expression for f-1(x), for x in the domain of f-1. = y(x + 2) = 12 yx + 2y = 12 = xy = 12 - 2y = 12 - 2y x= -- x= 12 - 2, and by switching x and y, we get y= 12 - 2 f(x) = 12 (d) Find the value or say "does not exist" (DNE). 1.84(f(n)) = 1 ii. f(p-10)) = DNE, because 0 is not in the domain of f-1

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.

3. For each limit below, evaluate it or determine that it does not exist. Use the oo or-o symbol where appropriate. 1934 + 3x2 +1 (a) lim [4 points) Ba| p2 - 6 - + 4e/ (b) 22 -1 lim- 2-1-0-22+1 [4 points) (c) lim ello) - We evaluate usi g the definition of 77-€ I-e derivative (A solution using l'Hopital's rule or with no justification will earn 0 points) 14 points)

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