f(x,y) = lxl^k*lyl^k/(x^2+y^2) is convergent, find min k

 Posts: 9
 Joined: Sat Oct 01, 2011 11:21 pm
f(x,y) = lxl^k*lyl^k/(x^2+y^2) is convergent, find min k
how to find the mininum of k to make f(x,y) = lxl^k*lyl^k/(x^2+y^2) convergent?
in practice book, there is a question which is similar but simple than this one, like xy/(x^2+y^2) is convergent or not, i know this is not convergent, as when keep y constant, if y=x, lim(x>0)=1/2, while if y=0, lim(x>0)=0, so it is not convergent.
what about this one? tks in advance!
in practice book, there is a question which is similar but simple than this one, like xy/(x^2+y^2) is convergent or not, i know this is not convergent, as when keep y constant, if y=x, lim(x>0)=1/2, while if y=0, lim(x>0)=0, so it is not convergent.
what about this one? tks in advance!
Re: f(x,y) = lxl^k*lyl^k/(x^2+y^2) is convergent, find min k
You mean continuous?
Any k > 1 will do. If you consider the straight line y = ax through the origin, then the limit of f(x,ax) as x >0 along this line is 0 as long as 2k > 2.
Any k > 1 will do. If you consider the straight line y = ax through the origin, then the limit of f(x,ax) as x >0 along this line is 0 as long as 2k > 2.

 Posts: 9
 Joined: Sat Oct 01, 2011 11:21 pm
Re: f(x,y) = lxl^k*lyl^k/(x^2+y^2) is convergent, find min k
thank you owlpride for your reply and sorry for my wrong info... yes i mean continuous... but when k=2/3 will make f(x,y) continuous, which is <1... am i right?owlpride wrote:You mean continuous?
Any k > 1 will do. If you consider the straight line y = ax through the origin, then the limit of f(x,ax) as x >0 along this line is 0 as long as 2k > 2.

 Posts: 61
 Joined: Sun Apr 04, 2010 1:08 pm
Re: f(x,y) = lxl^k*lyl^k/(x^2+y^2) is convergent, find min k
Nope. k=2/3 will make the limit at 0 not exist: If you go along y=x, the limit will be infinity, where as if you go along y=0, the limit will be zero (since the function is identically 0 along the x and y axis).

 Posts: 9
 Joined: Sat Oct 01, 2011 11:21 pm
Re: f(x,y) = lxl^k*lyl^k/(x^2+y^2) is convergent, find min k
oh... ok, I see, my mistake tks for pointing out!blitzer6266 wrote:Nope. k=2/3 will make the limit at 0 not exist: If you go along y=x, the limit will be infinity, where as if you go along y=0, the limit will be zero (since the function is identically 0 along the x and y axis).
Re: f(x,y) = lxl^k*lyl^k/(x^2+y^2) is convergent, find min k
I have a question.
For k = 1, lim x*y / (x^2+y^2) = 0 for both x and y approaching 0
For x=y, the limit is 1/2.
So the limit doens't exist. I get that.
Now, if we use polar coordinates, we get:
lim (r cos a)^k (r sin a)^k / r^2, as both, r and a go to 0
lim r^2k [sin (2a) / 2]^k / r^2 = lim r^(2k2) [sin (2a) / 2]^k
2k  2 = 0, the limit is 0 since lim sin (2a) / 2 = 0 as r, a both approach 0
2k  2 > 0, then both approach 0
So I get that f continuous if k >= 1
Can someone explain to me where the problem is? Thanks
For k = 1, lim x*y / (x^2+y^2) = 0 for both x and y approaching 0
For x=y, the limit is 1/2.
So the limit doens't exist. I get that.
Now, if we use polar coordinates, we get:
lim (r cos a)^k (r sin a)^k / r^2, as both, r and a go to 0
lim r^2k [sin (2a) / 2]^k / r^2 = lim r^(2k2) [sin (2a) / 2]^k
2k  2 = 0, the limit is 0 since lim sin (2a) / 2 = 0 as r, a both approach 0
2k  2 > 0, then both approach 0
So I get that f continuous if k >= 1
Can someone explain to me where the problem is? Thanks
Re: f(x,y) = lxl^k*lyl^k/(x^2+y^2) is convergent, find min k
DDswife: I could be wrong, but I believe your error is sending both r and a to 0. If you just send r to 0, you should get the limit at (0,0). By also sending a to 0, I think you're evaluating the limit by approaching (0,0) along the line y=0.
So, for k > 1, lim [r^(2k2) sin(2a)/2 , r>0] = 0. For k=1, lim [r^(2k2) sin(2a)/2 , r>0] = lim[sin(2a)/2, r>0] = sin(2a)/2.
Note that this agrees with your y=x example, because in that case a=pi/2, so sin(2a)/2=1/2.
Hope this helps!
So, for k > 1, lim [r^(2k2) sin(2a)/2 , r>0] = 0. For k=1, lim [r^(2k2) sin(2a)/2 , r>0] = lim[sin(2a)/2, r>0] = sin(2a)/2.
Note that this agrees with your y=x example, because in that case a=pi/2, so sin(2a)/2=1/2.
Hope this helps!
Re: f(x,y) = lxl^k*lyl^k/(x^2+y^2) is convergent, find min k
That might be the problem. Funny, when I took Multivariable Calculus, I watched some videos and learnt this trick and, if I am not confused, this is the way it was done. But I will recheck on that.
Thanks for your reply. BTW, I think that you meant a = pi/4, then 2a would be pi/2. But I got your point.
Good to know I am not by myself! This forum seemed to be abandonned lately. Are you planning to takie the MGRE at some point?
Thanks for your reply. BTW, I think that you meant a = pi/4, then 2a would be pi/2. But I got your point.
Good to know I am not by myself! This forum seemed to be abandonned lately. Are you planning to takie the MGRE at some point?
Re: f(x,y) = lxl^k*lyl^k/(x^2+y^2) is convergent, find min k
You were right. I just found this link and they say that r approaches 0 and the rest needs to be independent of the angle. Good that I corrected one of my mistakes! Thanks again.
http://www.math.ucla.edu/~skalyanswamy/ ... polar.pdf
http://www.math.ucla.edu/~skalyanswamy/ ... polar.pdf
Re: f(x,y) = lxl^k*lyl^k/(x^2+y^2) is convergent, find min k
Great! Glad I could help. Yeah, it should have said a=pi/4...I also left off an exponent inside the limit.
Yeah, the forum does seem pretty dead, but that could have to do with the time of year. I applied to grad schools a couple of years ago and the peak time for this forum was the OctoberApril stretch when the admission process is in full swing.
I am hoping to make another run at grad school this year; if I do, I'll take the MGRE in October. Are you also aiming for Fall 2015?
Yeah, the forum does seem pretty dead, but that could have to do with the time of year. I applied to grad schools a couple of years ago and the peak time for this forum was the OctoberApril stretch when the admission process is in full swing.
I am hoping to make another run at grad school this year; if I do, I'll take the MGRE in October. Are you also aiming for Fall 2015?
Re: f(x,y) = lxl^k*lyl^k/(x^2+y^2) is convergent, find min k
I am far away yet. Taking Linear Algebra next semester in college and, if the University finally accepts me (I am not from the US and they keep requesting stuff I don't even think I will be able to submit! But who know...), I will be taking Abstract Algebra. Watching some Topology videos on these days.
I am a Math teacher but my Abstract Algebra class was completely different from the classes here. So different, that I don't even understand the questions from the GRE! I didn't know what a symmetric group was till I googled for that. I didn't know what Z/Z2 was. I used the notation Z/R, were R is a relation, or Z_2, [0], etc. Now I see that these things are sort of the same. I didn't know what a coset was, either. My class and the ones I see you take here are as similar as a hand and a foot!
My Topology class wa lousy (half a year without a teacher to begin with!), and it was only about Metric Spaces. I didn't even know what a topology was!
And I need to take Complex Analysis. Never took this one.
But I am already preparing the GRe to take it at some point (whenever I think that I am finally ready!). And I will be around this forum.
I am a Math teacher but my Abstract Algebra class was completely different from the classes here. So different, that I don't even understand the questions from the GRE! I didn't know what a symmetric group was till I googled for that. I didn't know what Z/Z2 was. I used the notation Z/R, were R is a relation, or Z_2, [0], etc. Now I see that these things are sort of the same. I didn't know what a coset was, either. My class and the ones I see you take here are as similar as a hand and a foot!
My Topology class wa lousy (half a year without a teacher to begin with!), and it was only about Metric Spaces. I didn't even know what a topology was!
And I need to take Complex Analysis. Never took this one.
But I am already preparing the GRe to take it at some point (whenever I think that I am finally ready!). And I will be around this forum.
Re: f(x,y) = lxl^k*lyl^k/(x^2+y^2) is convergent, find min k
I have posted several messages lately with solutions to the exercises. If you find mistakes, or if you have better ways to solve them, let me know.