<s>[INST] Limiting Behavior of a Rational Function at Infinity Consider the following function:
$$f(x) = \frac{2x^2+x}{x^2-1}$$
I know there's a horizontal asymtote at $y = 2$. Without graphing, is there a way to tell that the $x\rightarrow \infty$ part is approaching from above and the $x \rightarrow -\infty$ is approaching from below?
 [/INST] For $x > 1$,
$$\frac{2x^2+x}{x^2-1} > \frac{2x^2+1}{x^2-1}>\frac{2x^2-2}{x^2-1}=2$$
Since the denominator is positive and $2x^2+x > 2x^2+1$. So $f(x) \to 2$ from above as $x \to \infty$.
Similarly
For $x< -2$, 
$$\frac{2x^2+x}{x^2-1} < \frac{2x^2-2}{x^2-1}=2$$
Since the denominator is positive and $2x^2+x < 2x^2-2$.
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