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Check if below limits exist $\lim_{(x,y,z) \to (0,0,0)} \left( {\frac{{2{x^2}y - x{z^2}}}{{y^2 - xz}}} \right).$ Is there any succession to prove that this limit is not zero?

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    @Teddy. OK, I'll take that into account, thanks.2012-07-24

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Taking the sequence $x_n = z_n = \frac{1}{n}$, and $y_n = \frac{1}{n} + \frac{1}{n^2}$, you get $ \lim_{n\to \infty} \frac{\frac{1}{n^3}+\frac{2}{n^4}}{\frac{2}{n^3}+\frac{1}{n^4}} = \frac{1}{2}.$ You can easily find a sequence for which the limit is $0$, and hence the limit does not exist.

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    wonderful solution. +12012-07-24