Growth rate of integration

Question

Let $f,g:[0,\infty)\to [0,\infty)$ where $\int_{0}^\infty g(x)dx<\infty$. Is it possible to find $f,g$ such that $$\frac{\int_0^a f(x)g(x)dx}{f(a)\int_a^\infty g(x)dx}$$ grows exponentially on $a$?

The best result I got is to take $f(x)=e^x$ and $g(x)=e^{-x}$ so the above equals to $a$, which growth linearly on $a$.

Answer 1

Take $$ f(x) \equiv 1, \; g(x) = e^{-x} $$ Then $$ \frac{\int_0^a f(x)g(x)dx}{f(a)\int_a^\infty g(x)dx} = \frac{\int_0^a e^{-x} \, dx}{\int_a^\infty e^{-x} \, dx} = \frac{1 - e^{-a}}{e^{-a}} = e^a - 1 $$ which is an exponential growth.