Homework 2 - Math 53

Math 53 - HOMEWORK # 2

( Deadline : December 4, 2009 )

Directions: Answer on a yellow pad. Show all solutions as necessary and BOX all Final Answers. 0.5 pt each )

1.) Given below is the graph of f x . Find the following limits, if they exist. If the limit is CN or KN, indicate so.

(a) lim f x

(c)

(b) xlim f x /4

(d) lim

x / 4K

2.5 pts each )

2 pts )

1.5 pts each )

lim

x /K4 C x /K4

f x

(e)

f x

(f)

lim

f x

(g) x /CN lim f x

lim

f x

(h) lim

x /K2 C x / 2K

x/2

f x

2.) Answer the following questions : 2

2x and then use them to sketch the graph. 2 x K1 x C1 (b) Determine the largest interval ( or union of intervals ) on which f x = is x K1 continuous. sin π$x (c) Suppose f x is defined on the open interval 0, 1 and f x = . Define x x K1 f x at 0 and 1 so that it is continuous on the closed interval 0, 1 . (a) Find the vertical asymptotes of f x =

Kx2

3.) Use the Intermediate Value Theorem to show that there is a root of the equation e in the interval 0, 1

=x

4.) Write TRUE if the statement is always TRUE. Otherwise, write FALSE. In any case, justify your answer. (a) A function can only have at most two horizontal asymptotes. (b) If f x O 1 for all x and lim f x exists, then lim f x O 1 x/0

x/0

(c) Let f x be any function which is continuous on the open interval a, b . Then d a function g x which is continuous on = and which satisfies g x = f x for all x 2 a, b 2 pts each )

4 points

5.) Find the limit of the following, if they exist x Kx sin x 3 K3 (a) lim (c) lim x / 0 2 x Ctan x x /CN 3x C3Kx 1 (b) lim x2$sin (d) lim 2 x C 4 x2 C3 x x/0 x x /KN 6.) Prove that if xlim f x = L, then xlim f x /a /a

TOTAL : 30 pts

= L