Expanding the limit, we get (x^2+2x∆x+∆x^2-2x-2∆x+1-x^2+2x-1)/∆x Crossing the 1s , the 2xs, and the x^2s out, we get (2x∆x+∆x^2-2∆x)/∆x Dividing the ∆x, we get  2x+∆x-2.  Making the limit of ∆x=0, we get 2x-2.