Question

determine the asymptotes of the function graphed below.
oa. there is a vertical asymptote at ( x = 5 ) and a horizontal asymptote at ( y = 3 ).
ob. there is a horizontal asymptote at ( x = 5 ) and a vertical asymptote at ( y = 3 ).
oc. there is a horizontal asymptote at ( y = 5 ) and a vertical asymptote at ( x = 3 ).
od. there is a vertical asymptote at ( y = 5 ) and a horizontal asymptote at ( x = 3 ).

Explanation:

Step1: Recall the definitions of vertical and horizontal asymptotes

A vertical asymptote is a vertical line \(x = a\) where the function approaches \(\pm\infty\) as \(x\) approaches \(a\). A horizontal asymptote is a horizontal line \(y = b\) which the function approaches as \(x\to\pm\infty\).

Step2: Analyze the graph

Looking at the graph, as \(x\) approaches \(3\) (from the left and right), the function values go to \(\pm\infty\), so \(x = 3\) is a vertical asymptote. As \(x\to\pm\infty\), the function approaches \(y = 5\), so \(y = 5\) is a horizontal asymptote.

Answer:

C. There is a horizontal asymptote at \(y = 5\) and a vertical asymptote at \(x = 3\).