Question

use long division to rewrite the rational function. what are the asymptotes of f? sketch the graph.
f(x)=\frac{4x}{x+6}

rewrite the function using long division.
f(x)=4-\frac{24}{x+6}

what are the horizontal asymptotes of f? select the correct choice below and, if necessary, fill in the answer box within your choice.
\bigcirc a. there is a horizontal asymptote defined by the line \square.
\quad (type an equation. type an exact answer, using radicals as needed.)
\bigcirc b. there is no horizontal asymptote.

Explanation:

Step1: Analyze the degrees of numerator and denominator

For the rational function \( f(x)=\frac{4x}{x + 6} \), the degree of the numerator (the highest power of \( x \)) is 1, and the degree of the denominator is also 1.

Step2: Apply the rule for horizontal asymptotes

When the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 4, and the leading coefficient of the denominator is 1. So the horizontal asymptote is \( y=\frac{4}{1}=4 \). Also, from the rewritten form \( f(x) = 4-\frac{24}{x + 6} \), as \( x\to\pm\infty \), the term \( \frac{24}{x + 6}\to0 \), so \( f(x)\to4 \).

Answer:

A. There is a horizontal asymptote defined by the line \( y = 4 \)