Integral of Gaussian curve

Question

I have a Gaus-like integrand. Could you please give me a clue how to integrate it for given constant $a,b>0$? $$\displaystyle \int\limits_{0}^{a} \lim \limits_{\epsilon \rightarrow 0}\,e^{\displaystyle \frac{1}{\epsilon}\left(ax-\frac{1}{2}bx^{2}\right)}dx$$

Answer 1

Hint: Assuming $\epsilon\to0^+$, since $\frac1\epsilon\to\infty$, if $ax-\frac12bx^2>0$, the integrand is $\infty$; if $ax-\frac12bx^2<0$, the integrand is $0$.