Question

1. (5pts) for the given function, find all vertical asymptotes and horizontal asymptotes, or state that there are none.

$f(x) = \frac{x^2 + 1}{2x^2 - 1}$
horizontal asymptote: _______________
vertical asymptote(s): _______________

Explanation:

Step1: Find horizontal asymptote

For rational functions, compare degrees of numerator and denominator. Both are degree 2. The horizontal asymptote is the ratio of leading coefficients:
$$y = \frac{1}{2}$$

Step2: Find vertical asymptotes

Set denominator equal to 0 and solve for $x$:
$$2x^2 - 1 = 0$$
$$2x^2 = 1$$
$$x^2 = \frac{1}{2}$$
$$x = \pm\frac{\sqrt{2}}{2}$$

Answer:

Horizontal Asymptote: $\boldsymbol{y=\frac{1}{2}}$
Vertical Asymptote(s): $\boldsymbol{x=\frac{\sqrt{2}}{2}}$ and $\boldsymbol{x=-\frac{\sqrt{2}}{2}}$