1) Let $f(t) = \frac{exp(-\sqrt{t})}{\sqrt{t}}, t>0$. Show that $\underset{t \rightarrow \infty}{lim} f(t) \sim O(exp(-\sqrt{t}))$.
2) Let $f(t) = exp(-1/t)\frac{1}{t^2}, t>0$. Show that $\underset{t \rightarrow \infty}{lim} f(t) \sim O(\frac{1}{t^2})$.
My understanding is that exponential function dominates polynomial terms whenever the order of polynomial is less than 1 (does this make sense?) but I do not know how to approach this formally. I appreciate any hints. The functions $f(t)$ pertain to the pdf of function of exponentially distributed random variable. This however is an appendage and does not help studying the tail behavior of functions.
In the first case the Bachmann–Landau notation $f(t)=O(\exp(-\sqrt{t}))$ means you have to show $$\limsup_{t\to\infty}\frac{|f(t)|}{\exp(-\sqrt{t})}<\infty.$$ Since $$\frac{|f(t)|}{\exp(-\sqrt{t})}=\frac{1}{\sqrt{t}}\overset{t\to\infty}{\to}0$$ the $\limsup$ is zero. The second part of your qestions works exactly like the first one, so there shouldn't any difficulies any more.