Question

find all horizontal asymptotes of the following function.

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

Explanation:

Step1: Divide numerator and denominator by \(x\)

$$\begin{align*} \lim_{x ightarrow\infty}f(x)&=\lim_{x ightarrow\infty}\frac{2(x + 7)}{2x-7}\\ &=\lim_{x ightarrow\infty}\frac{2(1+\frac{7}{x})}{2-\frac{7}{x}} \end{align*}$$

Step2: Evaluate the limit

As \(x
ightarrow\infty\), \(\lim_{x
ightarrow\infty}\frac{7}{x}=0\).
So \(\lim_{x
ightarrow\infty}\frac{2(1 + 0)}{2-0}=1\)

Also, \(\lim_{x
ightarrow-\infty}\frac{2(x + 7)}{2x-7}=\lim_{x
ightarrow-\infty}\frac{2(1+\frac{7}{x})}{2-\frac{7}{x}} = 1\)

Answer:

\(y = 1\)