To locate a vertical asymptote of a rational function, which procedure is described?

• A. Set numerator to zero
• B. Factor, cancel holes, then set bottom = 0
• C. Differentiate the function
• D. Set the derivative equal to zero

Answer: (B)

Vertical asymptotes occur where a rational function becomes unbounded, which happens where the denominator is zero after factoring and removing any common factors with the numerator. To locate them, factor both numerator and denominator, cancel any common factors (these cancellations create holes rather than asymptotes), and then set the remaining denominator equal to zero.

For example, f(x) = (x−2)/(x^2−4) factors to (x−2)/[(x−2)(x+2)]. Cancel the common factor (x−2) to get 1/(x+2) (for x ≠ 2). There’s a hole at x = 2, and a vertical asymptote at x = −2 where the simplified denominator is zero.

Setting the numerator to zero finds x-intercepts, not vertical asymptotes. Differentiating or setting the derivative to zero targets critical points, not asymptotes. If you skip canceling and just set the denominator to zero, you might falsely identify a hole as an asymptote.