Question

the function f is shown below. determine the equations of all vertical and horizontal asymptotes of f and the equivalent limit equations.
there are a total of zero vertical / horizontal asymptotes.

Explanation:

Step1: Recall the definition of vertical asymptote

A vertical asymptote occurs at \(x = a\) if \(\lim_{x
ightarrow a^{-}}f(x)=\pm\infty\) or \(\lim_{x
ightarrow a^{+}}f(x)=\pm\infty\). Looking at the graph of the function \(y = f(x)\), there is no value of \(x\) for which the function approaches \(\pm\infty\). So, there are no vertical asymptotes.

Step2: Recall the definition of horizontal asymptote

A horizontal asymptote occurs if \(\lim_{x
ightarrow\pm\infty}f(x)=L\), where \(L\) is a finite number. As \(x
ightarrow-\infty\), the function \(y = f(x)\) approaches a finite value. Let's assume \(\lim_{x
ightarrow-\infty}f(x)= 3\) (by observing the left - hand behavior of the graph). So, the equation of the horizontal asymptote is \(y = 3\) and the limit equation is \(\lim_{x
ightarrow-\infty}f(x)=3\)

Answer:

There are no vertical asymptotes. The horizontal asymptote is \(y = 3\) and the limit equation is \(\lim_{x
ightarrow-\infty}f(x)=3\)