Verified answer

lim (x-->∞) √(9x^2 + x) - 3x

= lim (x-->∞) √(9x^2 + x) - 3x * [√(9x^2 + x) + 3x]/[√(9x^2 + x) + 3x]

= lim (x-->∞) (9x^2 + x - 9x^2)/[√(9x^2 + x) + 3x]

= lim (x-->∞) x / [√(9x^2 + x) + 3x]

= lim (x-->∞) x / [x√(9 + 1/x) + 3x]

= lim (x-->∞) x / [x√(9 + 0) + 3x]

= lim (x-->∞) x / (3x + 3x)

= lim (x-->∞) x / 6x

= 1/6

I hope this helps!