Prove that $\int\limits_0^1 \frac{e^{-st}}{t^2}\,dt$ diverges.
My progress: I have encountered several problems so far where integral was improper and given integration was possible to express explicitly. However, my main struggles in this problem are $1)$ I can not even find to which function this is equivalent and $2)$ How can I use limit concept to prove the divergence? Any help or ideas would be strongly welcomed!
If $s\geq0$, $st\leq s$ for $0<t<1$ and so, $\frac{e^{-st}}{t^2}\geq \frac{e^{-s}}{t^2}$. From $\int^1_0\frac{1}{t^2}\,dt=\infty$ you get that the integral of interest diverges to infinity. A similar argument works if $s<0$.