a common method to solving these kind of limits is to divide the top and bottom of the fraction by the highest power of x in the denominator. since the highest power in the denominator is x (x^1), divide the top and bottom by x so that you get 2x^2 / (1 + 9/x). if you now take the limit as x approaches infinite, the 9/x goes to 0 and the 2x^2 goes to infinite. so you get infinite / 1 which is infinite which means the limit does not exist.