## College Algebra (6th Edition)

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). ------------------ $r(x)=\displaystyle \frac{x}{x^{2}+4}$ $p(x)=x\qquad$... fully factored $q(x)=x^{2}+4\qquad$... fully factored Nothing to reduce.... $q(x)=0$ $x^{2}+4=0$ $x^{2}=-4$ ... no real number has a negative square. Since the denominator , q(x) has no zeros r is defined for all real numbers, meaning Vertical asymptotes: none Holes: none