find all horizontal asymptotes of the following function.\n\n$f(x)=\\frac{2(x + 7)}{2x - 7}$

Explanation:

Step1: Divide numerator and denominator by (x)

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

Step2: Evaluate the limit

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

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

Answer:

(y = 1)