Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
(2) Let r1 1 and -(-) 1 (n+1)2 = I+ur (a) Show that lim,, T, exists. (b) Prove that z, #t1 by induction and find the limit.
1. Find lim(x,y)=(1,1) x2-y2 2xy 2. Show that lim(x,y)-(0,0) 21 z does not exist 3. Show that lim(x,y)=(0,0) z?”, does not exist 4. Find lim(x,y)=(0,0) eye if it exists, or show that the limit does not exist
1. Evaluate each of the following limits: (4 pts ea) b) lim 2- a) lim VF-1 bilim 7-x
5a) (5 pts) Find lim inf (xn) and lim sup (rn), for rn = 4 + (-1)" (1 - 2). Justify your answer 5b) (5 pts) Find a sequence r, with lim sup (xn) = 3 and lim inf (x,) = -2. 5c) (10 pts) Let {x,} be a bounded sequence of real numbers with lim inf (x,) = x and lim sup (x,) = y where , yER. Show that {xn} has subsequences {an} and {bn}, such that an...
1. (2 pts each) The graph of some unknown function f is given below. 10 6/ 8-64-2 624 10 12 Use the graph to estimate the following quantities: (0 f (9) (g) f(4) b) lim (a) lim (e) (d) lim ( 6) (e) lim f(x) (c) lim f(x) if g(x)f(x) 6) a value of r where f is continuous but not differentiable (k) a value of r where f"(x) 0 and f"(x)>0 (1) the location of a relative maximum value...
1. (10 pts) Evaluate limits. Apply L'Hopital's Rule when needed. (Inx) a. lim b. lim 1/(1-2) 20+
lim 00 lim 3a. (10 pts) Let Sn: 10,1] - R be defined by fn (x) = nºx (1 - )". Is is true that ( sn (2) de - 1 In (x) dx. 3b. (10 pts) Let fn (x) = 1+2+3* € (0,1). Find I In (2) de Justify your answer. lim
(10 pts) CMOS 1. (5 pts) Briefly explain how this circuit works and show a truth table. 2. (3 pts) Also, it has a flaw. Identify that flaw. 3. (2pts) Show how to alter the circuit to fix the flaw A B CF ini Bin IIn out -[ ( C' ㄧㄧㄧ inn