jmc

The numerator of the given function has degree $7$. Hence, if $q(x)$ has degree less than $7$, the function will have no horizontal asymptote. Therefore, the degree of $q(x)$ must be at least $7$. To see that $7$ works, we can consider $q(x)=x^7$. Then as $x$ grows away from $0$, the $x^7$ terms in the function will dominate and the function will tend towards $\frac{2x^7}{x^7}=2$. Thus the smallest possible degree of $q(x)$ is $\boxed{7}$.