MATH 180 Study Guide - Final Guide: Maxima And Minima, 4Dx, Asymptote

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turquoisegnu128

13 Dec 2018

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Problem 1 solution: compute each limit or explain why it does not exist. (a) lim x (b) lim x 3 cos2(x) 1 x . 4 + 3x + 5x2 (c) lim x + . Solution: (a) this limit is of the form 0/0. Thus, l"hopital"s rule applies. lim x cos2(x) 1 x . = lim x (cos2(x) 1) (x ) . = 0 (b) the function f (x) = |x 3|/(x 3) may be rewritten as f (x) = (cid:26) 1 if x > 3. Thus, the one-sided limits are lim x 3+ f (x) = 1, lim x 3 f (x) = 1. Since the one-sided limits are di erent, the limit does not exist. (c) this limit is of the form / . In fact, it must be applied twice. lim x + . Problem 2 solution: compute the derivative of each function below. Do not simplify your answer. (a) x9e2x (b) log5(4x + 2) (c) arcsin(1/x)

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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.

What are the answers for all of these questions? I am confused on how to solve for these.