Find $$\lim_{(x,y) \to (\infty,1)} \left(1 + \frac{1}{x}\right)^{\frac{x^2y}{x+y}}$$
Well, I know that $\lim_{x\rightarrow \infty}\left(1 + \dfrac{1}{x}\right)^x = e^1$
So I multipled the nominator and denominator of the exponent by $x$ so I can get closer to euler. but now I'm left to compute the limit of
${\dfrac{x^2y}{(x+y)x}} = {\dfrac{xy}{(x+y)}} = ?$
I tried polar, and I got $\dfrac{r\cos\phi \sin\phi}{\cos\phi + \sin\phi} = ?$
The question is, will the question mark be zero? because I know that the nominator is "zero". and I'm not sure about the denominator, I know that it is between $-2 \le x \le 2$. but if it is zero, I'm having a problem there. Maybe what I said is wrong and the denominator is just a number and makes altogether zero? $\frac{0}{k} = 0, k \in -2 \le x \le 2$ ?