2.2: Asymptotes and Holes | Precalculus

How it works

Factor numerator and denominator before describing a rational graph. Every original denominator zero is excluded. A canceled factor can produce a hole, while a remaining denominator zero typically produces a vertical asymptote after the expression is reduced. Cancellation does not restore an excluded input.

\frac{(x-a)p(x)}{(x-a)q(x)}=\frac{p(x)}{q(x)}\quad(x\ne a)

Worked examples

Example 1

Analyze $\displaystyle f(x)=(x^2-1)/(x-1)$ .

Factor and cancel to get $\displaystyle x+1$ , retaining $\displaystyle x\ne1$ . The graph is a line with a hole at $\displaystyle (1,2)$ .

Example 2

Find the vertical asymptote of $\displaystyle g(x)=(x+1)/(x-2)$ .

The denominator vanishes at $\displaystyle 2$ without cancellation, so $\displaystyle x=2$ is a vertical asymptote.

Problems

Problem 2.2.1

Simplify $\displaystyle (x^2-4)/(x-2)$ and state the hole.

Show worked solution

The reduced expression is $\displaystyle x+2$ , with $\displaystyle x\ne2$ . The hole is $\displaystyle (2,4)$ .

Problem 2.2.2

Find the excluded input and vertical asymptote of $\displaystyle 1/(x+3)$ .

Show worked solution

$\displaystyle x=-3$ is excluded and is a vertical asymptote.

Problem 2.2.3

For the graphed function $\displaystyle f(x)=(x^2-1)/((x-1)(x-2))$ , state the hole and vertical asymptote.

The graph reduces to (x + 1)/(x − 2), with a hole at (1, −2) and an asymptote at x = 2. — Original denominator exclusions remain after cancellation.

Show worked solution

Reducing gives $\displaystyle (x+1)/(x-2)$ , with $\displaystyle x\ne1,2$ . The hole is $\displaystyle (1,-2)$ and the vertical asymptote is $\displaystyle x=2$ .

The graph reduces to (x + 1)/(x − 2), with a hole at (1, −2) and an asymptote at x = 2. The hole is (1, −2); the vertical asymptote is x = 2. — The hole is (1, −2); the vertical asymptote is x = 2.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 2.2.1

Problem 2.2.2

Problem 2.2.3

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