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May 9th, 2014, 09:01 PM  #1 
Newbie Joined: Apr 2014 From: California Posts: 10 Thanks: 3  How do I prove this multivariable limit problem?
Compute $\displaystyle \lim_{x,y\to (0,0)}$ $\displaystyle 3xy\over x^2+y^3$ along lines of the form $\displaystyle y=mx$, for $\displaystyle m \neq 0$. What can you conclude? When I do it, I end up with 3m which is a limit. I know the function is discontinuous at (0,0) but I don't know how to prove it. 
May 9th, 2014, 09:26 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,655 Thanks: 2633 Math Focus: Mainly analysis and algebra  The thing about that answer is that you get a different limit for every direction of approach to (0, 0). When when dealing with a singlevariable limit, that means that the limit doesn't exist (or is a pair of onesided limits). Actually, I get $\frac{3m}{1 + m^2}$ (valid for all $m$) by that approach, but the problem I identified before still applies. 

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limit, multivariable, problem, prove 
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