Question

find all horizontal asymptotes of the following function.

$f(x)=\frac{2(x + 7)}{2x - 7}$

Explanation:

Step1: Simplify the function

First, expand the numerator: \(f(x)=\frac{2x + 14}{2x-7}\).

Step2: Use the rule for horizontal asymptotes of rational functions

For a rational function \(y=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}\), if \(n = m\) (where \(n\) is the degree of the numerator and \(m\) is the degree of the denominator), the horizontal asymptote is \(y=\frac{a_n}{b_m}\).
Here, the degree of the numerator \(n = 1\) (since the highest - power of \(x\) in \(2x + 14\) is \(x^1\)) and the degree of the denominator \(m = 1\) (since the highest - power of \(x\) in \(2x-7\) is \(x^1\)).
The leading coefficient of the numerator \(a_n=2\) and the leading coefficient of the denominator \(b_m = 2\).
So, \(\lim_{x
ightarrow\pm\infty}f(x)=\lim_{x
ightarrow\pm\infty}\frac{2x+14}{2x - 7}\). Divide both the numerator and denominator by \(x\): \(\lim_{x
ightarrow\pm\infty}\frac{2+\frac{14}{x}}{2-\frac{7}{x}}\).
As \(x
ightarrow\pm\infty\), \(\lim_{x
ightarrow\pm\infty}\frac{14}{x}=0\) and \(\lim_{x
ightarrow\pm\infty}\frac{7}{x}=0\). Then \(\lim_{x
ightarrow\pm\infty}\frac{2+\frac{14}{x}}{2-\frac{7}{x}}=\frac{2 + 0}{2-0}=1\).

Answer:

One Horizontal Asymptote (\(y = 1\))