This is problem 3.12 in Paul Garrett's 2019 example problems.
For $f \in \mathscr{S} $ (the space of Schwartz functions), show that $$\lim_{t \rightarrow + \infty} f(x) * \frac{2 \sin tx}{tx} = f(x) $$ ($*$ is the convolution operator).
I'm not sure in what sense the expression is supposed to converge (pointwise?).
I'd appreciate a hint about where to start; My first thought was to try showing that $2 \operatorname{sinc}(tx)$ are approximate identities, but their integrals are all $2 \pi$, so this approach quickly seemed like it might not work.