I'm pretty sure the answer is no, since it limits to 1, by L'hopital's rule, so it stays bounded. But a solution that I am comparing my work to claims that 0 is a singularity for sin(x)/x.
I am using dominated convergence theorem, and want to show that $e^{-\epsilon x} \frac{sin(x)}{x}$ is bounded above by $e^{-\epsilon x}$, on $[0,\infty)$ for $\epsilon >0$. This bound is also integrable on $[0,\infty)$.