Hello I am looking into a problem that I encountered, when going through fourier transform. If we take the definition of DFT for fourier series coefficients then it would be:
$$F(\omega)=\frac{1}{N}\sum_{n=0}^{N-1} x_n e^{i2\pi n/N}$$
Then when N would approach $\infty$ the whole formula could become a definition for the same principle for continuous signal, when swapping the sum with integral.
Now to my problem, I transformed a continuous sine with certain angle frequency $\omega_0$ obviously resulting in two dirac delta pulses in $\pm\omega_0$. Now when I would like to compute the coefficients I would need to "divide" the whole result by infinity, or rather compute something along those lines $$ \lim_{(N,\omega)\to(\infty,\omega_0)} \frac{\delta(\omega-\omega_0)}{N}$$ Now it would work perfectly afterwads, only if this limit was defined and the result would be 1, but I am unable to compute this limit or tell if it exists. I know the standart approach for acquiring coefficients for continuous signal (integrating over one period), but I was curous, if this approach can be valid as well. Thank you for your help in advance.