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I have trouble finding the bandwidth of a signal. Say I have an info bearing signal m(t)=sinc(2t/pi). I found the fourier transform of the sinc function and found that the angular frequency was 1/pi. I am confused whether the the bandwidth is w or 2w. Since it is band-limited, does that mean only the positive w is counted and the bandwidth is just 1/pi?

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When you look on a signal's DFT transform what you see is given in normalized units in the range $ [ -\pi, \pi ] $ where $ \pi $ equals the sampling rate.

Since the DFT is periodic and the way FFT work when one calculate the DFT in modern software (MATLAB / Python, etc...) the DFT is given in the $ [0, 2 \pi ] $ range.

In order to view it in the more intuitive way one could use fftshift() (MATLAB).

When you display it then, probably you won't see it gets to zero anywhere.
What you'll probably see (If the signal is band limited with bandwidth lower than the sampling rate) is a big drop in the magnitude.

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