Vertical Asymptotes

Problem 268: Let

$f(x) = \frac{1}{2x^2 - 5x - 12}.$

What is/are the vertical asymptote(s) of $f(x)$ ?

Solution: The vertical asymptotes of a rational function are given by the values of $x$ that makes the denominator zero. That is,

$2x^2 - 5x - 12 = 0 \implies (2x + 4)(x - 4) = 0 \implies x = -\frac{3}{2} \ \text{or} \ x = 4.$

This means that we have two vertical asymptotes which are given by the lines $x = 4$ and $x = -\frac{3}{2}$ .