Calculate limit (x^2+2x+1)/(2x^2-2x-1), x->+infinity

In order to calculate the limit of a rational function, when x tends to +inf., we'll divide both, numerator and denominator, by the highest power of x, which in this case is x^2.

We'll have:

lim[(x^2+2x+1)/(2x^2-2x-1)]= lim(x^2+2x+1)/lim(2x^2-2x-1)

lim x^2(1 + 2/x + 1/x^2)/lim x^2(2 - 2/x - 1/x^2)

After simplifying the similar terms, we'll get:

(lim 1 + lim 2/x + lim 1/x^2)/(lim 2 - lim 2/x - lim 1/x^2)

lim[(x^2+2x+1)/(2x^2-2x-1)]=(1+0+0)/(2-0-0)=1/2