I have been asked to provide an "approximation at infinity" of an expression that at the end simplifies to $\frac{b e^{a t}a e^{b t}}{ab}$, in a course about extreme value theory. In the course, we saw "approximations" such as $2 t^{\alpha }t^{2 \alpha }$ being approximated to $2 t^{\alpha }$ whatever the vague word approximation means. In words, this "approximation" states that the distribution tail is dominated by the term $2 t^{\alpha }$ at infinity. I think that there is no polynomial term $t^{\alpha }$ which dominates at infinity in the given question, since $e^{k t}$ decreases faster than any term $t^{\alpha }$. However, is there any sense in which this "approximation at infinity" can be taken? I tried to take the Taylor series at infinity and Mathematica returns the same expression (unevaluated?) and computing manually, the first order approximation of $e^{k t}$ is 0. Any ideas or references about how to compute these sort of approximations in extreme value theory are appreciated.
Do you have information on the relative sizes of $a$ and $b$? If we have, say $a>b$, then we can say that as $t \to \infty$, $\frac{b e^{a t}a e^{b t}}{ab}=e^{b t}(\frac{b e^{(ab) t}a }{ab}) \sim \frac{a e^{b t}}{ab} =\frac{e^{b t}}{1\frac{b}{a}}$. It is rather simple, but I don't think there's much more that you can say about it. 


If $\alpha > 0$, then you have $$ \lim_{t \to \infty} {2t^{\alpha}  t^{2\alpha} \over 2t^{\alpha}} = 1$$ and this is the sense in which such approximations are meant. Of course something stronger can be said  how quickly do these limits go to 1?  but usually when these approximations come without any qualifiers, this is what they mean. 


Maple says it this way... 

