I am tasked with evaluating $$\lim_{(x, y) \to (0, 0)} (1+x^2+y^2)^{\frac{1}{x^2+y^2+x y^2}}.$$
We can rewrite the expression it as follows:
$$ e^{\ln(1+x^2+y^2)\cdot\frac{1}{x^2+y^2+x y^2}} $$
Now, my first guess was that this tends to $1$. But that is apparently incorrect, it tends towards $e$, which one can see just by graphing it. That is also what the book says.
Now, we have the "standard limit":
$$\lim_{x \to 0^+} \frac{\ln(1+x)}x = 1$$
So I can kinda see how we can get the power to tend toward $1$ so we get $e^1$, but the expression is not exactly the same, we have $xy^2$ tacked on to the end of the denominator of the fraction. Furthermore, $xy^2$ doesn't have to be positive.
I tried using the substitution $x=\frac{1}{y^2}$, that gives us an expression that tends towards infinity both at $0$ and infinity and is never zero.
Can anyone point out to me what I'm missing. It feels like it should be super-obvious.