MATH 221 Midterm: MATH 221 UW Madison Exam1

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turquoisegnu128

31 Jan 2019

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5 points per problem: sketch the graph of the function g(x) = 2x 1 x 1 if if 1 < x < 2 if. Use the graph to state the value of each of the following limits, if it exists. (a) limx 1 g(x) (b) limx 1+ g(x) (c) limx 0 g(x) (d) limx 2 g(x) (e) limx 2+ g(x: evaluate the limit, if it exists. limx 2. 16 x 2 : state the de nition of f (x) is continuous at a. Use the de nition of con- tinuous and the properties of the limit to show that f (x) = 1 x+3 is continuous at x = 1: find the horizontal and vertical asymptotes of the curve y = 27 x3: find the equation of the tangent line to the curve y = x. 2 x at the point (0, 0).

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5. (22 pts) x² - 4 x-5 if x < 2, Let g(x)= sin (1 x) if x 22. (I) Evaluate the limit or state that it does not exist. If the limit does not exist, explain why. Show your work. (Note: Possible answers include + cor -.) (a) (3 pts) lim g(x) = ; (b) (3 pts) lim g (x) = (II) (2 pts) Determine whether g has any horizontal asymptotes. If so, write the equation (or equations) of all horizontal asymptotes. (III) (4 pts) Determine whether g has any slant asymptotes. If so, write the equation (or equations) of all slant asymptotes. (IV) (2 pts) Determine whether g has any vertical asymptotes. If so, write the equation (or equations) of all vertical asymptotes. (V) (8 pts) Determine whether g is continuous at 2. Use the definition of continuity, when justifying your answer. SHOW YOUR WORK.

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.

5. (22 pts) if x < 2, Let g(x) = sin (1 x) if x2 2. (I) Evaluate the limit or state that it does not exist. If the limit does not exist, explain why. Show your work. (Note: Possible answers include + cor -0.) (II) (2 pts) Determine whether g has any horizontal asymptotes. If so, write the equation (or equations) of all horizontal asymptotes. (III) (4 pts) Determine whether g has any slant asymptotes. If so, write the equation (or equations) of all slant asymptotes. (IV) (2 pts) Determine whether g has any vertical asymptotes. If so, write the equation (or equations) of all vertical asymptotes.

MATH 221 Midterm: MATH 221 UW Madison Exam1

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