Recent content by pantin

Homework Statement lim (x,y,z)->(0,0,0) (xy+yz+xz)/(x^2+y^2+z^2) lim(x,y) - > (1,1) (x-y^4)/(x^3-y^4) lim(x,y)->(0,0) [1-cos(x^2+y^2)]/(x^2+y^2)^2 one more.. is there a constant c in R s.t. the fn f(x,y) =( xy+y^3 ) / (x^2+y^2) for (x,y) =/= (0, 0) f(x,y) = c for (x,y)...

help! 'set' question Give an example in the set notation of a CLOSED set S in R^2 such that the closure of int S is not equal to S. I originally used the set s={ (x,y) : 0 <x^2+y^2<1} but I just noticed it's not closed set! ... can anyone give me an example? Thanks!

The thm says: Every bded monotone sequence in R is convergent. The proof: suppose {x_k} is a bded increasing sequence. Let M be the sup of the set of values {x_1, x_2,...} I claim that x_k -> M. Since M is an upper bd, we have x_k <= M for all k. (***) on the other hand, since M is...

so /a_n+1 - a_n/ < epislon is one of the case of /a_k - a_j/ < epislon right. but it cannot stand for all the cases

very clear! (meaning when is 'large enough'), all the terms of the sequence will be 'very close' to that number. that's the explanation I want! thx a lot statdad! thanks boombaby!

why totally different?? i thought they were in the same pattern x_k-x_j->0 as k,j -> infinity, that is if for every epislon>0 there exists an integer K s.t. /x_k - x_j/<epislon whenever k>K and j >K

I get it , thanks statdad. but I have a question on your explanation above., how would you explain the second sentence on the graph. i mean , like the proof of least upper bd M, x > M - epislon for any epislon>0 in this epislon proof, i understand the meaning , which is, no matter how small...

Homework Statement Let a_n = (1/2)[(1/a_n)+1] and a_1=1, does this sequence converge? Homework Equations A sequence in R^n is convergernt if and only if it's cauchy. A sequence in R^n is called a cauchy sequence if x_k - x_j ->0 as k, j-> infinity. The Attempt at a Solution I...

let me try to plug in some number to the fn in the solution tomorrow...too late tonight, going to sleep.. thanks for asking :)

you are right, but i still don't 100% get it. my professor gave me a similar example, f(x)=c. as you said, x can be any number, assume S={x_k}, f(S)=c, right? but if a set is made up by the points on a line, say y=c, then this is not compact because it's not closed? i am not sure...

suppose f:R^m -> R^n is a map such that for any compact set K in R^n, the preimage set f^(-1) (K)={x in R^m: f(x) in K} is compact, is f necessary continuous? justify. The answer is no. given a counterexample, function f:R->R f(x):= log/x/ if x is not equal to 0 f(x):= 0 if x=0...

i got it, it doesn't contain 0 , which is the limit of the sequence thx a lot

oh yes! haha, you are smart, it doesn't say the line segments do not include endpoints,that's just the coordinate sign! but then how to continue..

we just have 2 similar concepts on the text, one is every bded sequence in R has a convergent subsequence the second is what I stated before.. every bded sequence in R^n has a convergent subsequence or should I use the definition of upper and lower bd?

Homework Statement Consider the function f:R^2->R defined by f(x,y)=[e^(x+y)]-y+x. Is there an absolute maximum value of f on the set s={(x,y):/x/+/y/<=2}? Justify. note, /x/ is the absolute value of x. Homework Equations a. If f is con't, it takes compact sets to compact sets...