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Assuming that k is real is much weaker than assuming that k is an integer. Also it seems that you need to simplify the expression first to let the computation work:
sage: N = var('N')
sage: k = var('k')
sage: assume(k-1>0)
sage: assume(k, 'integer')
sage: E = (6*N^k*(N-1)^(1-k))/((2*N-1)*(k+1))
sage: limit(E.full_simplify(),N=oo)
3/(k + 1)
