## College Algebra (6th Edition)

Vertical asymptote: $x=-4$ Hole at $x=0$
Locating Vertical Asymptotes, (page 395) tells us that for $f(x)=\displaystyle \frac{p(x)}{q(x)}$, if p(x) and q(x) have NO common factors, and a is a zero of the denominator $q(x)$ , then the line $x=a$ is a vertical asymptote. BUT, if they DO have common factors, after REDUCING the form of the function's equation, the number a may not cause the denominator to be zero any more. in which case there will be a hole at x=a. The point is that both p(x) and q(x) have to be factored, and if we can, then we reduce the expression for f(x). ------------------ $h(x)=\displaystyle \frac{x}{x(x+4)}$ $p(x)=x\qquad$... fully factored $q(x)=x(x+4)\qquad$... fully factored We can reduce the expression for h(x): $h(x)=\displaystyle \frac{x}{x(x+4)}=\frac{1}{x+4}, \quad x\neq 0$ the denominator is zero for $x=-4,$ h(x) is undefined for $x=0$ Vertical asymptote: $x=-4$ Hole at $x=0$