Equation: x^2 - 64 / ((3x+4)(x-5))

The horizontal asymptote is 1/3. I graphed this and for very large values of x, the graph equals appromixmately .354 and a little up. How is 1/3 an asymptote then? My books ays to make x equal a very large positive number, and then it will be (x^2)/(3x^(2)), which is 1/3. Am I looking at the graph wrongly?