For the integral: $$I=\int^{t/2}_{-t/2} e^{i(\omega_1-\omega_2)t'}dt'$$ What would be the lower limit on $t$ for the approximation: $$I\approx 2\pi \delta(\omega_1-\omega_2)$$ to hold?
I would guess (and this is only a guess) that it would be: $$t>>\frac{1}{\min(\omega_1,\omega_2)}$$ If this is correct how would we show it, and if not what is the limit on $t$?