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Given the function $f(x,y) = \frac{xy}{x+y}$, after my analysis I concluded that the limit at $(0,0)$ does not exists.

In short, if we approach to $(0,0)$ through the parabola $y = -x^2 -x$ and $y = x^2 - x$ we find that $f(x,y)$ approaches to $1$ and $-1$ respectively. Therefore the limit does not exists.

I think my rationale is right. What do you think?

Alternatively, is there another approach for this problem?

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    Your approach is correct. The limit must be independent of the path to the origin.2012-03-14
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    $f(x,y)$ can not be determine at $(0,0)$2012-03-14
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    Note that your paths are tangent at the origin to the line $y=-x$, which is not in the function's domain. When looking for paths for limit calculations like this, finding such tangent paths is one of the strategies. (Another basic strategy, which is not helpful in this problem, is to try $y=kx$.)2012-03-18
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    @alex.jordan A selection of a few examples where tangent paths are the way to go would make for a nice post on the site.2018-03-19

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