Lambert W function: limits and inverse relation

Question

I'm looking for a specific limit of a function containing The lambert W function.

for the most simplified case, my function looks as follows:

$y=W(-rxe^x)$

this is the plot of the function for $r=0.1$:

As you can see from the figure, $y\rightarrow -\infty$ somewhere $x\approx1.1$.

I would like to find this limit.

When I Inverse the function using mathematica, I get a solution (which I think is only one branch) of the form:

$x=W(-\frac{1}{r} y e^y)$

The mathematica plot of that function looks as follows:

As you can see from the figure, when $y\rightarrow\infty$, then $x\rightarrow0$.

This does not coincide with the previous result.

Can anyone please, reference, direct or explain how to solve the following problems:

1. Finding the inverse function of $y(x)=W(-rxe^x)$, i.e, $x(y)$.

2. Find the value of $x$ in which $y(x)=W(-rxe^x)$ converges to, in the limit of $y(x)\rightarrow -\infty$.

Answer 1

1) The branch point $x_b$ of LambertW is at $x_b=-\frac{1}{e}$. So your limit point $x_l$ is the solution of $ -\frac{1}{e}= -rxe^x$, and therefore $$x_l(r) = W\left(\frac{1}{re}\right)$$

$$x_l(0.1) = W\left(\frac{1}{0.1e}\right)=W\left(\frac{10}{e}\right) = 1.156868396615\dots$$

The limit is not $-\infty$ but $$\lim_{x\to x_l^{-}}y(x) = -1$$ For $x > x_l$ the function values $y(x)$ are complex.

2) The inverse of $y(x) = W(-rxe^x)$ is indeed $$x(y) = W\left(-\frac{ye^y}{r}\right)$$ and with Maple I get limit(LambertW(-1/r*y*exp(y)), y=-infinity) = 0.

But note that $x(-1) = x_l,$ so there is no inconsistency to $y(x_l) = -1.$