protip: http://www.wolframalpha.com/input/?i=lim+h-%3E0+%28%282%28x%2Bh%29%2F%28x%2Bh%2B1%29%5E2%29+-+%282x%2F%28x%2B1%29%5E2%29%29%2Fh

lim_(h->0) ((2 (h+x))/(h+x+1)^2-(2 x)/(x+1)^2)/h
Factor the numerator and denominator:
= lim_(h->0) (h (-2 (h x+x^2-1)))/((((x+1)^2 (h+x+1)^2) (h ((x+1)^2 (h+x+1)^2)))/((x+1)^2 (h+x+1)^2))
Cancel terms, assuming h/((x+1)^2 (h+x+1)^2)=!=0:
= lim_(h->0) -(2 (h x+x^2-1))/((x+1)^2 (h+x+1)^2)
Factor out constants:
= -(2 (lim_(h->0) (x^2+h x-1)/(h+x+1)^2))/(x+1)^2
The limit of (h x+x^2-1)/(h+x+1)^2 as h approaches 0 is (x-1)/(x+1):
= -(2 (x-1))/(x+1)^3