Note that the phase zero reference point of an FT is where the cosine or real component of all the exponential basis vectors is 1.0, and where the sine or imaginary component is not only zero, but has a 1st derivative of 1.0. This only occurs in a DFT or FFT at sample 0 of all basis vectors from 0 to N-1.
At the center (or N/2 of N even), the 1st derivative of the sine or imaginary component flips from -1.0 to 1.0 between basis vectors (crosses zero in the opposite directions for odd periodic and even periodic basis exponentials). So that does not meet the criteria for being the phase zero reference of an FT.
Thus the need for an fftshift (for even N).
This works because all the DFT basis vectors are circular, thus any rotation of the input data just results in a shift to needed phase reference.
As for clipping the limits from -N/2 to N/2 instead of -inf to inf: if the area under the curve from N/2 to inf is on the order of or less than the numerical noise (quantization, rounding, etc.) then you might not even notice the difference after printing or plotting to some readable number of significant digits.