jmc

To determine the horizontal asymptotes, we consider what happens as $x$ becomes very large. It appears that, as $x$ becomes very large, the rational function becomes more and more like $$y\approx \frac{5x^2}{3x^2},$$so it should become closer and closer to $\frac{5}{3}$.

We can see this explicitly by dividing both the numerator and denominator by $x^2$. This gives $$y=\frac{5-\frac{9}{x^2}}{3+\frac{5}{x}+\frac{2}{x^2}}.$$Indeed, as $x$ gets larger, all of the terms other than 5 in the numerator and 3 in the denominator become very small, so the horizontal asymptote is $y=\boxed{\frac{5}{3}}$.