Question

the function f is shown below. determine the equations of all vertical and horizontal asymptotes of f and the equivalent limit equations.

Explanation:

Step1: Identify vertical asymptotes

Vertical asymptotes occur where the function approaches infinity or negative - infinity. From the graph, the function approaches infinity or negative - infinity at $x=- 1$ and $x = 6$. So the equations of vertical asymptotes are $x=-1$ and $x = 6$. The equivalent limit equations are $\lim_{x
ightarrow - 1^{-}}f(x)=\pm\infty$, $\lim_{x
ightarrow - 1^{+}}f(x)=\pm\infty$, $\lim_{x
ightarrow6^{-}}f(x)=\pm\infty$, $\lim_{x
ightarrow6^{+}}f(x)=\pm\infty$.

Step2: Identify horizontal asymptotes

As $x
ightarrow\pm\infty$, the function approaches $y = 0$. So the equation of the horizontal asymptote is $y = 0$. The equivalent limit equations are $\lim_{x
ightarrow-\infty}f(x)=0$ and $\lim_{x
ightarrow\infty}f(x)=0$.

Step3: Count the asymptotes

There are 2 vertical asymptotes ($x=-1$ and $x = 6$) and 1 horizontal asymptote ($y = 0$), so in total there are 3 asymptotes.

Answer:

three