Mathfolis

Rational Functions and Vertical Asymptotes

Unit 1 · Polynomial and Rational Functions

What AP Precalc asks here

A vertical asymptote (VA) of r(x)=p(x)/q(x)r(x) = p(x) / q(x) is a value cc where the denominator vanishes but the numerator does not. Near the VA, ∣r(x)∣→∞|r(x)| \to \infty. The multiplicity of the factor in the denominator controls whether the two one-sided limits agree: odd multiplicity → opposite infinities, even multiplicity → same infinity on both sides.

Vertical asymptote condition

VA at x=cx = c
q(c)=0  AND  p(c)≠0q(c) = 0 \;\text{AND}\; p(c) \neq 0

One-sided behavior near x=cx = c

Odd multiplicity in denominator
r→+∞ on one side, −∞ on the otherr \to +\infty \text{ on one side, } -\infty \text{ on the other}
Even multiplicity in denominator
r→same infinity on both sidesr \to \text{same infinity on both sides}
Caution: If the numerator also vanishes at the same factor, the point is a hole, not a VA. Always compare numerator and denominator multiplicities before classifying.
Type 1

Locate vertical asymptotes

Factor the denominator, list its roots, and discard any that also kill the numerator. What remains gives the VAs.

Example 1
Find all vertical asymptotes of r(x)=x+1(x−2)(x+3)r(x) = \dfrac{x + 1}{(x - 2)(x + 3)}. (A) x=−1x = -1 only (B) x=2x = 2 and x=−3x = -3 (C) x=2x = 2, x=−3x = -3, and x=−1x = -1 (D) No vertical asymptotes

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Type 2

Distinguish vertical asymptote from hole

Identify the shared factor; cancel it. If the denominator still vanishes at the candidate after cancellation, it's a VA; otherwise it's a hole.

Example 2
For r(x)=(x−1)(x+2)(x−1)(x−4)r(x) = \dfrac{(x - 1)(x + 2)}{(x - 1)(x - 4)}, classify x=1x = 1 and x=4x = 4. (A) x=1x = 1 is a VA; x=4x = 4 is a hole (B) x=1x = 1 is a hole; x=4x = 4 is a VA (C) Both are VAs (D) Both are holes

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Type 3

One-sided behavior at a VA

Test the sign of numerator and denominator just to the left and just to the right of the VA. The sign of the quotient there tells you which infinity rr tends to.

Example 3
For r(x)=1x−2r(x) = \dfrac{1}{x - 2}, what are the one-sided limits at x=2x = 2? (A) −∞-\infty from both sides (B) +∞+\infty from both sides (C) −∞-\infty from the left and +∞+\infty from the right (D) +∞+\infty from the left and −∞-\infty from the right

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Rational Functions and Vertical Asymptotes | AP Precalculus — Mathfolis