Folded sine approximation to a Dirac delta ($m\sin (m|t|)\to 2{\delta }_0$)

As distributions on the line, $m\sin (m|t|)\to 2{\delta }_0$. Pairing with a test function $A$ folds the integral to ${\int }_0^{\infty }m\sin (ms)[A(s)+A(-s)]ds$. One integration by parts yields $2A(0)$ plus a cosine integral tending to zero by the Riemann-Lebesgue lemma. Pointwise convergence is unnecessary.