lim
h->0 [h - sqrt(x+h) + sqrt(x)]/h
The key to this is the binomial theorem:
(x+h)^n = x^n + n*x^(n-1)*h + n(n-1)/2 * x^(n-2) * h^2 + ...
So, we find ourselves with
(h + sqrt(x) - sqrt(x+h))/h
= (h + x^(1/2) - (x^(1/2) + 1/2 * x^(-1/2)*h + <higher powers of h>)/h
= 1 - 1/2 * x^(-1/2)
(all terms with h go to zero)