MATH 53 Sample Long Exam 1EAC Arances December 3, 2010

DIRECTIONS: Do as indicated. Show the necessary solution steps. Use only the right pages of yourbluebooks for your solution steps and answers. Use the left pages for scratch work. Box all final answerswhen necessary.

I. TRUE or FALSE. Write true if the statement is always true. Otherwise, write false.

1. If limxa+

f(x) and limxa

f(x) both exist, then the limit of f(x) as x a exists.2. Let f be a function defined on the closed interval [a, b] such that f(a) < 0 and f(b) > 0,

then f has a zero in the interval (a, b).

3. If limxaf(x) = 0 and g(x) is a function defined at x = a, then limxa

(f

g

)(x) = 0.

4. If limx+f(x) = 0 and limx+g(x) = +, then limx+(fg)(x) = 0.

5. There is a function, not identically zero, that is both odd and even.

II. LIMIT EVALUATION. Answer completely. Make sure to use proper limit notation.

1. limx pi

12+

1 tan 3x1 cosx

2. limx7

54 + 3x|7 x|

3. limx

(3x2 + 8x+ 6

3x2 + 3x+ 1

)

4. limx2

x 2tan pi2x

5. limx1

x3 x2 10x 8x3 + 5x2 + 7x+ 3

6. limx

sin2 3x

x2

III. DO AS INDICATED.Given the function f , defined as

f(x) =

2 xx+ 4

x < 2[[x+ 4]] 2 x < 0x2 + 3x

x+ x20 x

1. Identify all possible points of discontinuity of f .

2. Test the continuity of f at each of those points. If a discontinuity occurs, classify it.

IV. Given the system of equations {y = 6 sinx

y = 4 (2x)3 ,

use the intermediate value theorem to show that the system has a solution in x in the interval(0, pi6

).

END OF EXAM!Total: points