Zeros, Vertical Asymptotes, and Holes

Factor first, then read the graph off the factors

Write $r(x)=\dfrac{p(x)}{q(x)}$ in fully factored form. Every linear factor now falls into one of three roles:

• Zero: a factor of the numerator only. $r(a)=0$ and the graph meets the $x$-axis at $x=a$. If its multiplicity is odd the graph crosses; if even it touches and turns.
• Vertical asymptote: a factor of the denominator only. As $x\to a$, $|r(x)|\to\infty$; the two one-sided limits are each $+\infty$ or $-\infty$, decided by a sign check on either side.
• Hole (removable discontinuity): a factor shared by numerator and denominator, with numerator multiplicity $\ge$ denominator multiplicity. The limit exists there; its value — plug $x=a$ into the reduced form — gives the coordinates of the hole. If the denominator's multiplicity is larger, the factor still produces a vertical asymptote after partial cancellation.

Multiplicity also controls asymptote behavior: at a vertical asymptote from an even power, both one-sided limits share the same sign ($+\infty$ both sides or $-\infty$ both sides); from an odd power, they are opposite.