This equality holds for *all* real numbers N,M 
Can you see why?pic.twitter.com/LvR2r0y6iT
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I don't get how 1/z for z=0 is taken as -> ∞ . It could be - ∞ with equal validity.
The integral bounds are "really" 0+ to infinity, so when we make the substitution 1/x we take lim_{x--> 0+} 1/x
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A small + would do the trick . But, the original integral didn't show 0+. Hence the confusion.
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Right, but the + is assumed since we are integrating on the positive half of the real line
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