MATH 111 Midterm: MATH 111 NJIT M111 Exam1SP14

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turquoisegnu128

15 Feb 2019

School

New Jersey Institute of Technology

Department

Mathematical Sciences

Course

MATH 111

Professor

All

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Document Summary

All work must be shown in order to receive full credit. The use of books, notes, calcu- lators, or any other external sources of information is not allowed during this examination. If the limit does not exist, explain why: lim x 3. 2x 6 x2 2x 3: lim x 0 tan(3x) sin(2x) If the limit does not exist, explain why. b. lim x 0+ If the limit does not exist, explain why: lim x 2 x 2. 2 x + 2 b. lim x arcsin x2 + 7. Evaluate the following limits, allowing + and as possible values of a limit. If the limit does not exist, explain why. a. lim x 4 . X 2 b. x 0+ 1 x lim. Find the slope of the tangent line as the limit of a di erence quotient for f (x) = at x = 1. Find all horizontal, vertical, and slant (oblique) asymptotes, if they exist, for the following functions.

Related Questions

3. (21 pts) Evaluate the limit or say that the limit does not exist. If a limit does not exist, explain why. Show your work. You may NOT use a table of values, a graph, or L'Hospitals's Rule to justify your answer. (Note: Possible answers include + or -..) (a) (7 pts) x2 - 2x - 3 - lim x+3 Vx+1 - 2 x2 - 2x - 3 Vx+ 1 + 2 lim x+3 Vx+1 -2 Vx+1 +2 (x - 3) (x + 1) (Vx+ = lim x+3 x + 1 - 4 = lim x+3 (x - 3) (x + 1) (Vx+1 +2). -- x-3 = lin (x + 1) (Vx+1 +2) = (3 + 2) (V3+1 +2) = = 4 (V4 +2) = 16 (b) (7 pts) lim xsin (In x) X >0+ NOTE: We cannot apply the Product Law, since lim sin (1n x) DOES NOT EXIST. On the other hand, -1s sin (in x) s 1 and , therefore, -xs x sin (in x) sx, (since x > 0). Since lim X = lim (-X) = 0, x0+ X0 the inequality above implies that lim xsin (in x) = 0, by the Squeeze Theorem . x0+ EXPLANATION : Since lim ln x = -0, x →0+ sin (lnx) oscillates between 1 and -1 as x + 0*, and, therefore lim sin (In x) DOES NOT EXIST. x >0+ (c) (7 pts) in x+2- (lnx) +In 20 -= -00 (x - 2) 70-

4. (26 pts) Evaluate the limit or say that the limit does not exist . If a limit does not exist, explain why. Show your work. You may NOT use a table of values, a graph, or L' Hospitals's Rule to justify your answer. (Note: Possible answers include + or -.) X-2 (a) (10 pts) lim – x+2 782 +5 - 3 (b) (8 pts) xvot COS X OSCILLATES lim x → x + 4 x2 Since -1 scos x si, we have that -1 (c) (8 pts) cos x x + 4 x2 1 x + 4 x? x + 4 x2 -1 1 and since lim – Elim- x+ x + 4 x2 + x + 4 x2 * +422=0, it follows by the Squeeze Theorem that lim Co2 = 0. X +0 x + 4 x 2

MATH 111 Midterm: MATH 111 NJIT M111 Exam1SP14

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