<s>[INST] Calculate limit of $(1 + x^2 + y^2)^\frac{1}{x^2 + y^2 + xy^2}$ as $(x,y) \rightarrow (0,0)$ I'm trying to calculate the limit of 
$(1 + x^2 + y^2)^\frac{1}{x^2 + y^2 + xy^2}$ as $(x,y) \rightarrow (0,0)$.
I know that the limit is supposed to be $e$, and I can arrive at this answer if I study the univariate limit by, for instance, setting $x=t, y=0$ and letting $t \rightarrow 0$ or vice versa.
I'm not sure how to calculate it for the multivariate case. I tried using polar coordinates $(x= r \cos{\theta}, y = r \sin{\theta})$ which gets me
$$ (1 + r^2)^{\frac{1}{r^2(1 + r \cos{\theta}\sin^2{\theta})}} $$, after symplifying the expression, but I don't know how to proceed from there. I assume that I am supposed to end up with an expression similar to $(1 + n)^\frac{1}{n} \rightarrow e$ as $n \rightarrow 0$. 
 [/INST] 
I thought it might be instructive to present an approach that circumvents use of polar coordinates and relies instead on a straightforward application of the AM-GM inequality.  To that end we proceed.



To begin, the AM-GM inequality guarantees that for $x>-1$
$$\begin{align}
\left|\frac{xy^2}{x^2+(1+x)y^2}\right|&\le \frac{|x|y^2}{2|x||y|\sqrt{1+x}}\\\\
&=\frac{|y|}{2\sqrt{1+x}}
\end{align}$$
Hence, applying the squeeze theorem reveals
$$\lim_{(x,y)\to(0,0)}\left(\frac{xy^2}{x^2+(1+x)y^2}\right)=0\tag1$$


Next, we can write 
$$\frac{x^2+y^2}{x^2+y^2+xy^2}=1-\frac{xy^2}{x^2+(1+x)y^2}$$
Appealing to $(1)$, we have
$$\lim_{(x,y)\to(0,0)}\left(\frac{x^2+y^2}{x^2+y^2+xy^2}\right)=1\tag2$$


Finally, equipped with $(2)$, we find that
$$\begin{align}
\lim_{(x,y)\to(0,0)}\left(\left(\left(1+x^2+y^2\right)^{\frac1{x^2+y^2}}\right)^{\frac{x^2+y^2}{x^2+y^2+xy^2}}\right)&=\left(\lim_{(x,y)\to(0,0)}\left(\left(1+x^2+y^2\right)^{\frac1{x^2+y^2}}\right)^{\lim_{(x,y)\to(0,0)}\left(\frac{x^2+y^2}{x^2+y^2+xy^2}\right)}\right)\\\\
&=e^1\\\\
&=e
\end{align}$$
And we are done!
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