Calculating limits of a function of 2 or 3 variables I have to calculate these two limits, and have no idea where to start from.
Your guidance for how should I start working with it can help me a lot.
1) $\lim\limits_{(x,y,z)\rightarrow (0,0,0)} (1+xyz)^{(x^2+y^2+z^2)^{-1}}$
2)$\lim\limits_{(x,y)\rightarrow (0,0)} \dfrac{4y^2+3xy^2+2x^2}{x^2+2y^2}$
Thanks in advance.
 A: Hint:


*

*Write 
$$
\lim_{(x,y,z)\to (0,0,0)} (1+xyz)^{(x^2+y^2+z^2)^{-1}} = \lim_{(x,y,z)\to (0,0,0)} \left[(1+xyz)^{\frac{1}{xyz}}\right]^{\frac{xyz}{x^2+y^2+z^2}}
$$
The limit of the base behaves like $\lim\limits_{t\to 0}(1+t)^{\frac{1}{t}}$, while for $\lim\limits_{(x,y,z)\to (0,0,0)} {\frac{xyz}{x^2+y^2+z^2}}$, the top has degree three, while the bottom has only degree two, hence the top should be approaching $0$ in a higher order than the bottom, and the limit $\lim\limits_{(x,y,z)\to (0,0,0)} {\frac{xyz}{x^2+y^2+z^2}}$ should be $0$(this step is left for you to argue in a more mathematical way), and the whole limit should be 1. EDIT: you mention you haven't learned polar/spherical coordinates(that would be the easier way), here we could make use of AMGM inequality to get:
$$
\lim\limits_{(x,y,z)\to (0,0,0)} {\frac{|xyz|}{x^2+y^2+z^2}} \leq \lim\limits_{(x,y,z)\to (0,0,0)}{\frac{|xyz|}{3\sqrt[3]{x^2 y^2 z^2}}}
$$ 
and evaluating the limit for the right hand side is not that difficult.

*Write
$$
\lim\limits_{(x,y)\to (0,0)} \dfrac{4y^2+3xy^2+2x^2}{x^2+2y^2} = 2 + \lim\limits_{(x,y)\to (0,0)}\frac{3xy^2}{x^2+2y^2}
$$
like Didier said in the comment, now the argument is kinda similar, the top again is a one degree higher polynomial than the bottom(use AMGM inequality again for the bottom).
