Limit of Lambert $W$ Product Log is the Natural Log? In solving this equation $\large  y=x^ne^x$ I get the result that 
$$n \cdot W\left( \frac{y^{1/n}}{n}\right)=x $$
So now it is apparent to me that when $n=0$ you would simply get $\ln(y)=x$ by normal methods. But is there any way to show that the limit this is true as well?
$$\lim_{n \rightarrow 0}  n \cdot W\left( \frac{y^{1/n}}{n}\right)= \ln y$$

On any compact interval of $\mathbb{R}^+$ the sequence of continuous and monotonic functions given by 
$$ f_m(x) = x^{1/m} e^{x} $$
converges uniformly towards $e^x$, hence, for a given $y\in\mathbb{R}^+$, the sequence $\{x_m\}_{m\in\mathbb{N}^+}$ of the solutions of $f_m(x)=y$ converges towards the solution of $e^x = y$, i.e. $\log y$.
