pr.probability - Extreme value theory

I have been asked to provide an "approximation at infinity" of an expression that at the end simplifies to $-frac{b e^{-a t}-a e^{-b t}}{a-b}$, 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.