Im following Mark Thomson's Modern Particle Physics section 2.3.6 but I have question in the proceedure followed from equation 2.46 to 2.48
Starting from equation 2.46:
$$d \Gamma_{fi} = \frac{1}{T} \lvert T_{fi} \rvert^2 \int_{-T/2}^{T/2} \int_{-T/2}^{T/2} e^{i(E_f - E_i)t}e^{-i(E_f - E_i)t'} dt dt'.\tag{2.46}$$
The book says the solution to the double integral is sin$^2 x$/$x^2$ where $x = (E_f - E_i)T/2\hbar$ so the transition rate is only significant when $E_f \approx E_i$. This is something I fully agree.
But then it says that, as a consequence of the previous explanation we can re-write 2.46 as:
$$d \Gamma_{fi} = \lvert T_{fi} \rvert^2 \lim_{T\to\infty} { \frac{1}{T} \int_{-T/2}^{T/2} \int_{-T/2}^{T/2} e^{i(E_f - E_i)t} e^{-i(E_f - E_i)t'} dt dt' }.$$
But honestly I dont understand why. With such limit the result of the integrals (sin$^2 x$/$x^2$) is going to zero while $E_f - E_i$ is the quantity that should be close to zero.
Furthermore, the $\lim_{T\to\infty}$ is not needed. In my opinion the proceedure up to equation 2.49 does not require this limit. Am I right?
