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May 9th, 2014, 09:01 PM   #1
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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.
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May 9th, 2014, 09:26 PM   #2
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Originally Posted by RedBarchetta View Post
I end up with 3m
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 single-variable limit, that means that the limit doesn't exist (or is a pair of one-sided 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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