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

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=3). So the equations of vertical asymptotes are (x=-1) and (x = 3). The limit equations are (\lim_{x\rightarrow - 1^{-}}f(x)=-\infty), (\lim_{x\rightarrow - 1^{+}}f(x)=\infty), (\lim_{x\rightarrow3^{-}}f(x)=\infty), (\lim_{x\rightarrow3^{+}}f(x)=\infty).

Step2: Identify horizontal asymptotes

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

Answer:

Vertical asymptotes: (x=-1), (x = 3); Limit equations for vertical asymptotes: (\lim_{x\rightarrow - 1^{-}}f(x)=-\infty), (\lim_{x\rightarrow - 1^{+}}f(x)=\infty), (\lim_{x\rightarrow3^{-}}f(x)=\infty), (\lim_{x\rightarrow3^{+}}f(x)=\infty); Horizontal asymptote: (y = 0); Limit equations for horizontal asymptote: (\lim_{x\rightarrow-\infty}f(x)=0), (\lim_{x\rightarrow\infty}f(x)=0)