From Creighton and Anderson Chapter 7 on Gravitational Wave data analysis:
Please explain where the delta t and the limit came from in this derivation.
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Marcus is correct, so this means something else is wrong. The left equality simply shows that, for continuous time, the unilateral PSD, $S_x(f)$, is two times the Fourier transform of the autocorrelation function, $R_x(\tau)$. See, e.g., A.B. Carlson, Introduction to Communication Systems, 2nd Ed., McGraw-Hill, NY ©1975, Chapter 2, equation 18a. Or see pretty much any other book on Fourier transforms.
In discrete time, with consecutive samples spaced by $\Delta t$, the unilateral PSD for white noise is $2\sigma^2\Delta t$. Small values of $\Delta t$ are usually desirable, since the sampling frequency is the reciprocal of $\Delta t$ and the Nyquist frequency is half the sampling frequency. But $\Delta t$ cannot go to zero, as Marcus commented.
What happens if you just delete the limit stuff, on the right hand side of equation 7.22, and see what happens in the textbook's subsequent matched filter treatment? For a matched filter, the maximum signal-to-noise ratio is simply twice the pulse energy divided by the unilateral (white noise) PSD. (Same reference as above, Chapter 4, equation 12a.) So it looks like $2\sigma^2\Delta t$ will be used in your textbook. Hope this helps a bit.