Calculus Asymptote Questions

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Math 200 Name: Test #1 Form A . 1. For the function f whose graph is given, find the following. (20 pts) a. lim f ( x) = . x → b. lim f ( x) = . . x →− c. lim+ f ( x ) = . d. lim+ f ( x) = e. lim f ( x ) = . f. lim f ( x ) = x →−1 x →0 x→2 g. lim f ( x) = . x→2 . x →1 h. Vertical asymptote(s) is /are i. Horizontal asymptote(s) is/are . . j. Is f ( x ) continuous at x = 2 ? Explain your reasoning. 2. Which of the following statements are true, and which are false? Put answer in the answer blank provided below.(12pts) e x if x  0 a. lim+ f ( x) = 1 , if f ( x) =   x →0 ln x if x  0  e x if x  0 lim f ( x ) = 1 f ( x ) = b. , if   x → 0− ln x if x  0  f ( x) c. If lim f ( x) = 0 and lim g ( x) = 0 , then lim does not exist. x →c x →c x →c g ( x ) d. If f ( x ) is undefined at x = c , then the limit of f ( x ) as x approaches c does not exist. e. If f (c ) = L , then lim f ( x) = L. x →c f. If f ( x) = sin x , then lim+ f ( x) =  x→ Form A  2 In Exercises 3-6 circle the best answer. (16pts) 3. If f ( x) = x 2 − 9 , then what is lim ( f ( x) + x ) ? x →−2 a. −7 b. −3 c. 3 d. 7 e. The limit does not exist.  x2 − 1  if x  1  4. Let f ( x ) =  x − 1  Which of the following statements, I, II, III, are true?  2 if x = 1  I. lim f ( x) exists. x →1 a. only I b. only II II. f (1) exists. c. I and III d. I, II and III 5. Find value(s) of x at which the function f ( x) = a. x = 2 b. x = −2 6. Given the function f ( x) = a.) y = 0 c. x = 2 III. f ( x ) is continuous at x = 1 e. I and II ( x − 2)2 has a vertical asymptote when, x2 − 4 d. x = 0 e. none of these 10 x3 , f ( x ) has a horizontal asymptote when 2 x2 − 6 b.) y = 5 7. A line x = −10 is a if c.) y = 10 d.) y = 2 e.) Does not exist asymptote for the graph of a function f ( x) , . (6pts) 8. Sketch a possible graph of a function f ( x) , with asymptotes, satisfying all the following conditions. (6pts) f (−2) = −1, f (0) = 0, f (2) = 1, f (4) = −1, lim− f ( x) = , lim+ f ( x) = −, lim f ( x) = −2, x→3 Form A x→3 x→− f (6) = 3, lim f ( x) = , x→ 9. Find the exact value of each limit by algebraic methods. Do not give any decimal approximation (30 pts) Credit will only be given for work shown. 4( x − 6) x + 10 a. lim x →6 4 − x + 10 b. 1 1 − lim 2 + sin x 2 x →0 sin x c. x3 − 8 lim 2 x →2 x − 4 d. x4 − x2 + x + 6 lim x→− 3x5 − x + 5 e. lim x→− 5 − 2x 8 x 2 − 3x 10. Find the intervals on which the function x3 − 9 x 2 h( x ) = 3 x − 5x2 + 6 x Credit will only be given for work shown. 11. Find all the asymptotes of g ( x) = x3 − 27 . (6 pts) 2 x3 − 18 x Credit will only be given for work shown. Form A is continuous. (6 pts)
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