Suppose we have some sequences of positive real numbers $a_n$, $b_n$, with $a_n \to a \in \mathbb{R}$, $b_n \to \infty$.
Furthermore, suppose some constant $c \in \mathbb{R}$.
How could I determine the limit (if it exists) of $$ W_0 (a_n b_n \text{exp}(b_n +c)) - (b_n +c)$$,
where $W_0(x)$ is Lambert's W, defined as the real inverse of $f(x) = x \text{exp}(x)$.
I do not know which technique one could apply here. I tried to use some of the identities that one can find here, but did not find anything to useful.
For example, using $ \ln W_0(x) = \ln x - W_0(x)$, one could rewrite this as
$$\ln \frac{a_n b_n}{W_0 (a_n b_n \text{exp}(b_n +c))} $$
Any hints would be appreciated! Thank you.
On the wikipedia page of the Lambert W function, you can find the following useful approximation of $W_0$: $$W_0(x) = \ln(x) - \ln(\ln(x)) + o(1)$$ This means $$\lim_{x\to\infty}W_0(x) - (\ln(x) - \ln(\ln(x))) = 0$$ Because the argument of $W_0$ in your expression tends to infinity, we can substitute the approximation without changing the limit.
We can then compute as follows (using $\ln(a_nb_ne^{b_n+c}) = \ln(a_nb_n)+b_n+c$): $$ \begin{aligned} \lim_{n\to\infty} W_0 (a_n b_n e^{b_n +c}) - (b_n +c) &= \lim_{n\to\infty} \ln(a_nb_n) - \ln(\ln(a_nb_n)+b_n+c)\\ &=\lim_{n\to\infty} \ln\left(\frac{a_nb_n}{\ln(a_nb_n)+b_n+c}\right)\\ &=\lim_{n\to\infty} \ln\left(\frac{a_n}{\frac{\ln(a_nb_n)}{b_n}+1+\frac{c}{b_n}}\right)\\ &=\ln\left(\frac{\lim_{n\to\infty}a_n}{\lim_{n\to\infty}\frac{\ln(a_nb_n)}{b_n}+1+\lim_{n\to\infty}\frac{c}{b_n}}\right)\\ &=\ln\left(\frac{a}{0+1+0}\right)\\ &= \ln(a) \end{aligned} $$