Question

find all horizontal asymptotes of the following function.

$f(x)=\frac{2(x - 4)}{x + 5}$

Explanation:

Step1: Recall the rule for horizontal asymptotes of rational functions

For a rational function \(f(x)=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0}\), if \(n = m\), the horizontal asymptote is \(y=\frac{a_n}{b_m}\).

Step2: Identify the degrees of the numerator and denominator

The numerator \(2(x - 4)=2x-8\) is a first - degree polynomial (\(n = 1\)), and the denominator \(x + 5\) is also a first - degree polynomial (\(m=1\)).

Step3: Find the leading coefficients

The leading coefficient of the numerator \(a_n = 2\), and the leading coefficient of the denominator \(b_m=1\).

Step4: Calculate the horizontal asymptote

Using the formula \(y=\frac{a_n}{b_m}\), we have \(y=\frac{2}{1}=2\).

Answer:

The horizontal asymptote is \(y = 2\).