Rational function and horizontal asymptote

Question

Let $f(x)$ be a rational function of the form $\dfrac{p(x)}{q(x)}$, where $q(x) \neq 0$. Assume that $q(x)$ has two distinct roots $x_{1}$ and $x_{2}$ where $x_{1} < x_{2}$, and the degree of $p(x)$ is less that the degree of $q(x)$. If $\displaystyle \lim_{x \to x_{1}^{+}}f(x) = \lim_{x \to x_{2}^{-}}f(x)$, is it guaranteed that $f(x)$ will not pass through the horizontal asymptote?

Answer 1

What do you think of this one ?$$\frac{x^2-1}{x^4-2 }$$