I think I have it. Since we are talking about the complex plane we better have a definition of the zeta function in a neighborhood around s = 1. So from mathworld, a series representation is:

So now the above becomes:

From wikipedia we have that

and for z = 2 we have

Hence now we have

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I factored out the sum from the limit since at its limiting value it is just the logarithm of 2 so it does not affect the limit. By L'Hopital's rule the limit is:

Multiplying things out you get the desired expression.