Rational Functions: Asymptotes, Holes, and What to Do at Each

The cancellation rule (most important!)

For a rational function R(x) = p(x)/q(x), fully factor both numerator and denominator. Then:

• Factor that CANCELS → HOLE at that x-value
• Factor in denominator that does NOT cancel → VERTICAL ASYMPTOTE there

Example: R(x) = (x² − 4)/(x − 2) = (x − 2)(x + 2)/(x − 2) → cancels to (x + 2) with restriction x ≠ 2. So there is a HOLE at x = 2 (not an asymptote). At the hole, y = 2 + 2 = 4 — the point (2, 4) is removed.

Horizontal asymptote rules

For R(x) = p(x)/q(x):

• deg(p) < deg(q): y = 0
• deg(p) = deg(q): y = (leading coeff p)/(leading coeff q)
• deg(p) > deg(q): NO horizontal asymptote (oblique if exactly 1 more)

Slant (oblique) asymptote

When deg(p) = deg(q) + 1, perform polynomial long division: R(x) = quotient + remainder/q(x). The quotient is the slant asymptote; the remainder vanishes as |x| → ∞.

Check yourself

Practice with real CBE questions

Pre-Calc Sem A practice for rational function drills.