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! 



Upper limit function for the decay of $\int_x^\infty f(y)dy$, provided $f(y)$ is analytic, when $x\rightarrow\infty$?
$f(x):=\frac{\sin^2 x}{x^2}$ is entire, bounded on $\mathbb R$, and $\int\_x^\infty f(y)\ dy$ decreases as $cx^{-1}$.


