# Nasty Limit including Lambert-$W$ Function.

I would like to calculate the following limit:

$$A=\lim_{d\to 0^+}\exp\left[ -\left(\frac{d}{1-q}\right)\left(W_{0}\left[ B\left( 1+\frac{x}{rq}\right)^{\frac{1}{d}} \right]-W_{0}[B]\right)\right]$$ where $q,B,r>0$, $x \in \mathbb{R}$ and $W_0$ is the $k=0$ branch of the Lambert-$W$ function defined as: $$x=W(x)e^{W(x)}$$ Mathematica says that the limit is equal to: $$A=\left( 1+\frac{x}{rq} \right)^{\frac{1}{q-1}}$$ which seems correct to me, since I do expect power laws.

But I would really like to be able to evaluate this limit on my own. I mean, Mathematica will not show me how to proceed in order to reach the result. I have made some attempts but this limit is a nasty one.

Is it possible that someone who has some experience with this kind of limits helps me out a bit?

Thank you!

• Do you mean $d\to0^+$? – joriki Jun 26 '16 at 23:27
• @joriki Yes that would be correct. I will fix that, thank you. – Mitscaype Jun 26 '16 at 23:29

The term $W_0[B]$ doesn't contribute, since the corresponding factor goes to $1$ with $d\to0$. For the other term, use
$$W_0(x)=\log x-\log\log x+o(1)$$
(see Wikipedia). The $o(1)$ term doesn't contribute, so the limit is
$$$$\lim_{d\to 0^+}\exp\left[ -\left(\frac{d}{1-q}\right)W_{0}\left[ B\left( 1+\frac{x}{rq}\right)^{\frac{1}{d}} \right]\right]\\ = \lim_{d\to 0^+}\exp\left[ -\left(\frac{d}{1-q}\right)\left(\log\left(B\left( 1+\frac{x}{rq}\right)^\frac1d\right)-\log\log\left(B\left( 1+\frac{x}{rq}\right)^\frac1d\right)\right)\right]\\ = \lim_{d\to 0^+}\exp\left[-\left(\frac{d}{1-q}\right)\left(\log B+\frac1d\log\left(1+\frac{x}{rq}\right)-\log\left(\log B+\frac1d\log\left(1+\frac{x}{rq}\right)\right)\right)\right]\\ =\exp\left[\frac1{q-1}\log\left(1+\frac x{rq}\right)\right] \\ =\left(1+\frac x{rq}\right)^{\frac1{q-1}}\;.$$$$