I have the following limit:
$\lim_{x\rightarrow\infty}\left(1+\frac{a}{x^{1/2+\epsilon}}\left(1-\exp\left(-\frac{b}{x^{1/2+\epsilon}}\right)\right)\ln\left(\frac{a}{x^{1/2+\epsilon}}\right)\right)^x$
where $0.
I care for the case where $\epsilon>-1/2$. I suspect that for $\epsilon>0$ this limit evaluates to 1, and for $-1/2<\epsilon\leq0$ it evaluates to 0. However, I am having hard time evaluating this. I have tried taking the log of the expression (moving the $x$ in the exponent down), substituting $y=1/x$ and then Taylor-expanding the log, but didn't get anywhere.
Does anyone have any tips/hints that might help me evaluate this?