Limit $\lim \limits_{z\rightarrow 0^+} z \int_1^\infty x^2 e^{-zx^2-zx} \mathrm dx $

Question

Consider the following limit: $$\lim \limits_{z\rightarrow 0^+} z \int_1^\infty x^2 e^{-zx^2-zx} dx $$ Can we find the answer to this limit without calculating the integral?

I'll be thankful if you help me on this issue?

Answer 1

You can see it diverges if you change variables to $y= \sqrt{z} (x + \frac{1}{2})$, the limit you want becomes:

$$ \lim_{z\to 0^+} ze^{z/4}\int_{3z/2}^{\infty} \left(\frac{y}{\sqrt{z}}-\frac{1}{2}\right)^2 e^{-y^2} \frac{dy}{\sqrt{z}} $$ The problematic term is the one containing $y^2$ expanding the polynomial, because this has an overall $\frac{1}{\sqrt{z}}$, all other things give finite numbers in the limit, bu this factor makes the whole thing diverge.