Lesson · Holes versus vertical asymptotes

Holes versus vertical asymptotes

Publisher: Anannt EducationReviewer: Precalculus Academic Lead (designated)Last review: 10 September 2026~36 minFeatured complete lesson

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Prerequisite warning — not a lock

Recommended before this lesson: Factoring and invalid operations, Polynomial zeros and end behaviour. You may continue anyway (for example, to prepare school homework).

Retrieve

You will need difference of squares, a quadratic factorisation, and the fact that a quotient is undefined where the denominator is zero.

exactsymbolicfoundationalFamily fam-diff-squaresNo calculator

Factor

x^2-1

Answer

Exact form required on this item.

mcqsymbolicfoundationalFamily fam-rational-domainNo calculator

\dfrac{x+1}{x-1}

is undefined when

x=1

x=-1

x=0

never

exactsymbolicfoundationalFamily fam-factor-quadNo calculator

Factor

x^2-x-2

Answer

Exact form required on this item.

Editorial blocks

Why this works

A rational function is a quotient of polynomials. After factoring, common factors cancel *as algebraic products*, which is valid wherever those factors are nonzero. The original formula is still undefined wherever the original denominator was zero. If a cancelled factor was the only reason the denominator vanished, nearby outputs approach a finite height (a hole). If a factor remains in the denominator, outputs become unbounded (a vertical asymptote).

Common mistake

Treating every denominator zero as a vertical asymptote; dropping

x\neq a

after cancelling

(x-a)

; or classifying a hole correctly but using

0/0

as if it were a

y

-coordinate of

0

Check your result

Substitute a number near the excluded

x

into the original and the simplified expressions; they should match. At a hole, nearby outputs stay near the hole height. At a vertical asymptote, nearby outputs become large in magnitude.

Explain your thinking

Name the cancelled factors, list every original restriction, then classify each excluded

x

as hole or asymptote with a coordinate or equation.

Exam application

AP Precalculus Unit 1 tests rational functions across equation, table and graph. A graph-to-equation item later in review is more honest evidence than repeating the same factoring exercise.

Compact summary

Factor first. A cancelled factor that zeroed the denominator typically produces a hole; a surviving factor produces a vertical asymptote. Simplification never restores excluded inputs. For

h(x)=(x^2-4)/(x^2-x-2)

: simplifies to

(x+2)/(x+1)

, exclusions

x=2

and

x=-1

; hole at

(2,4/3)

; vertical asymptote

x=-1

Calculator guidance: No calculator is required. A graphing calculator may hide a hole at the pixel scale; factoring decides the discontinuity type. If you graph, use a table of values near the excluded input and compare with the algebra.

Exit to lessons

Next public lesson: Factoring and invalid operations

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