I am having difficulty seeing why this integral reduces to $1$ or $0$, just like the delta function. In the following statement, $\Delta$ is small, whereas $R$ goes to infinity:
$$\lim_{R\rightarrow \infty}\frac{2}{\pi}\int\limits^{k_{0}+\Delta}_{k_{0}-\Delta} k\ dk\ \left\{ \frac{\sin[(K-k)R]}{K^{2}-k^{2}} \right\}= \begin{cases} 1; & |K-k_{0}|<\Delta\\ 0, & |K-k_{0}|>\Delta \end{cases} $$
Context:
I came across this result at the bottom of p.765 of Morse & Feshbach's book, Methods of Theoretical Physics. The authors were trying to normalise eigenfunctions that took the form of Bessel functions and, after taking their asymptotic form, the statement reduced to this integral and they gave its result (above), which conforms with an earlier point they mentioned that such normalising integral will behave like the integral defining the delta function:
$$ \int\limits^{\Delta}_{-\Delta} \delta(z-\zeta)\ d\zeta= \begin{cases} 1; & |z|<\Delta\\ 0, & |z|>\Delta \end{cases} $$
I have tried typical methods to manipulate the integral, like integration by parts, etc, but couldn't get to the final answer above. Any suggestions?