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How can I show formally whether the following limit exists or not? $$\lim_{(x, y, z) \to (0, 0, 0)} \frac{x^2+2y^2+3z^2}{x^2+y^2+z^2}$$

I have tried to justify the non-existence of the limit by approaching with two curves such that the result is different, so the limit will not be unique:

Letting $f(x,y,z)$ be the function in the limit, I have that

$$\lim_{z \to 0} f(0, 0, z) = \lim_{z \to 0} \frac{3z^2}{z^2} = 3$$ and $$\lim_{x \to 0} f(x, 0, 0) = \lim_{x \to 0} \frac{x^2}{x^2} = 1$$ which are not the same.

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  • 1
    $\begingroup$ Yes, your method is correct! $\endgroup$ – Kenny Wong May 6 '17 at 22:23
  • $\begingroup$ You could try to improve your Latex. $\endgroup$ – hamam_Abdallah May 6 '17 at 22:25
  • $\begingroup$ If $x=ka, y=kb, z=kc $, limit = 6 $\endgroup$ – Takahiro Waki May 7 '17 at 3:45

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