Determine whether the limit exist, if so find its value : \begin{equation*} \lim_{(x,y) \rightarrow (0,0)} \frac{3}{x^2 + 2 y^2} \end{equation*}

Will the limit does not exist and its answer is $\infty$?

up vote 6 down vote accepted

The answer is $\infty$: $$\lim_{(x,y) \rightarrow (0,0)}\frac{3}{x^2 + 2 y^2} = \lim_{x \to 0}\left(\lim_{y \to 0}\frac{3}{x^2 + 2 y^2}\right) = \lim_{x \to 0}\left(\frac{3}{x^2}\right) = +\infty$$ Here is a plot of the function is terms of $(x,y)$. WolframAlpha shows that in the origin it blows up to infinity.

You can say that the limit exists, non existing limits are a different category, for example: $$\lim_{x \to 0}\frac{1}{x} = \pm \infty$$ depending on which values you're heading from ($\lim_{x \to 0-}$ or $\lim_{x \to 0+}$), the solution will be different.

Where $z=x+i\sqrt{2}y = re^{i\phi}$,

$$\lim_{z\to0} \frac{3}{|z|^2} = \lim_{r\to0} \frac{3}{r^2} = \infty$$

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