Find $\lim_{x\to \infty} \ln(\exp(\operatorname{LmW}(x))+1)(\exp(\operatorname{LmW}(x))+1) - x - \ln(x)$ Find $\lim_{x\to \infty} \ln(e^{\operatorname{LambertW}(x)}+1)(e^{\operatorname{LambertW}(x)}+1) - x - \ln(x)$
Where the $LambertW$ function is defined here : http://en.wikipedia.org/wiki/Lambert_W
How to do this ?
 A: Maple says
$$
\lim_{x \to \infty}\left[\operatorname{e} ^{\bigl(LambertW (x) + 1\bigr)} \operatorname{ln} \biggl(\operatorname{e} ^{\bigl(LambertW (x) + 1\bigr)}\biggr) - x - \operatorname{ln} (x)\right] = \infty
$$
if that's what you mean.  In fact,
$$
\lim_{x \to \infty}\frac{1}{x}\left[\operatorname{e} ^{\bigl(LambertW (x) + 1\bigr)} \operatorname{ln} \biggl(\operatorname{e} ^{\bigl(LambertW (x) + 1\bigr)}\biggr) - x - \operatorname{ln} (x)\right] = e-1
$$  
edit Oct 4 
OK, on this limit
$$
\lim_{x\to\infty}\left[\Bigl(\operatorname{e} ^{W (x)} + 1\Bigr) \operatorname{ln} \Bigl(\operatorname{e} ^{W (x)} + 1\Bigr) - x - \operatorname{ln} (x)\right]
$$
Maple says "Too many levels of recursion".  So I used $e^{W(x)}=x/W(x)$ and then let $x=e^y$.  (Also use: $\ln W(e^y) = y - W(e^y)$.)  Now Maple says
$$
\lim_{y\to\infty} \;\left[\frac{\operatorname{e} ^{y} \operatorname{ln} \bigl(\operatorname{e} ^{y} + W \bigl(\operatorname{e} ^{y}\bigr)\bigr)}{W \bigl(\operatorname{e} ^{y}\bigr)} - \frac{\operatorname{e} ^{y} y}{W \bigl(\operatorname{e} ^{y}\bigr)} + 
\operatorname{ln} \Bigl(\operatorname{e} ^{y} + W \bigl(\operatorname{e} ^{y}\bigr)\Bigr) - 2 y + W \bigl(\operatorname{e} ^{y}\bigr)\right] = -\infty
$$
which is claimed as your answer...
