1.6: End Behavior of Rational Functions | AP Precalculus

How it works

Compare numerator and denominator degrees to determine rational end behavior. A smaller numerator degree gives horizontal asymptote y = 0. Equal degrees give the ratio of leading coefficients. A numerator degree exactly one larger can give a slant asymptote found by division.

\frac{ax^n+\cdots}{bx^n+\cdots}\longrightarrow\frac ab\quad(|x|\to\infty)

Worked examples

Example 1

Find the horizontal asymptote of $\displaystyle (3x^2+1)/(2x^2-5)$ .

The degrees are equal, so the asymptote is the leading-coefficient ratio: $\displaystyle y=3/2$ .

Example 2

Find the slant asymptote of $\displaystyle (x^2+1)/(x-1)$ .

Division gives $\displaystyle x+1+2/(x-1)$ . The last term approaches zero, so the slant asymptote is $\displaystyle y=x+1$ .

Problems

Problem 1.6.1

Find the horizontal asymptote of $\displaystyle (2x+1)/(x^2+4)$ .

Show worked solution

The numerator degree is smaller, so the horizontal asymptote is $\displaystyle y=0$ .

Problem 1.6.2

Find the horizontal asymptote of $\displaystyle (5x^3-1)/(2x^3+x)$ .

Show worked solution

Equal degrees give $\displaystyle y=5/2$ .

Problem 1.6.3

Use the graph of $\displaystyle y=(x^2+1)/(x-1)$ to identify its slant and vertical asymptotes.

The graph of (x squared + 1)/(x − 1) approaches the dashed line y = x + 1 at its ends. — Compare the curve with the dashed line away from the vertical asymptote.

Show worked solution

Division gives $\displaystyle y=x+1+2/(x-1)$ . The slant asymptote is $\displaystyle y=x+1$ ; the vertical asymptote is $\displaystyle x=1$ .

The graph of (x squared + 1)/(x − 1) approaches the dashed line y = x + 1 at its ends. The asymptotes are x = 1 and y = x + 1. — The asymptotes are x = 1 and y = x + 1.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 1.6.1

Problem 1.6.2

Problem 1.6.3

Need help with this topic?