Question

evaluate lim as x approaches infinity of sqrt(9x^2 + x) - 3x Answer: 1/6 How?

Answer 1

Ok. So try forcing a rationalization first. We get lim x->inf x/(sqrt(9x^2+x)+3x)=lim x->inf 1/(sqrt(9x^2+x)/x+3)=lim x->inf 1/(sqrt(9x^2+x/x^2)+3). the limit of the inside expression can be obtained by L'Hopital's, since direct evaluation yields inf/inf. Taking the derivative of the top and bottom gives limx->inf (18x+1)/2x=lim x->inf 9+1/2x=9. Thus, we have the lim x->inf 1/(sqrt(9x^2+x/x^2)+3)=lim x->inf 1/(sqrt(9)+3)=1/6