Question with three parts:

a. Is there a theorem which states an upper limit function for rate of the decay of $\int_x^\infty f(y)dy$, as $x\rightarrow\infty$?

b. Assume $f(y)$ is positive, real, and restriceted to the positive-real y's, bounded from above by some constant 'a' and $\lim_{y\rightarrow\infty}[f(y)]\rightarrow0$ and is analytic in $[0,a]$:

Is it always possible to say something like e.g., $\int_x^\infty f(y)dy\leq exp(-cx)$, when $x\rightarrow\infty$?

c. If the answer to a.&b. is "nope". What are the general properties this integral in this limit?

Thanks on advance!