So I'm trying to find the asymptotic expansion as $x \to \infty$ of $$f(x)=\frac{1}{\bigg[A-\int \frac{\lambda^x}{\Gamma(x+1)}dx\bigg]^\frac{1}{\alpha}}$$
Note that $\lambda>0$ and $\alpha>0$. We can also see that $x \to \infty$ means that we can basically treat A as some arbitrary constant of integration and ignore the constant factor that comes out of the integral, but I have no clue on where to go from here. I'm basically stuck. Note here that $f(x) \to \infty$ is a constraint as well.