Question

give the equations of any vertical, horizontal, or oblique asymptotes for the graph of the rational function $f(x)=\frac{5x + 5}{x - 2}$. select the correct answer below and, if necessary, fill in the answer box to complete your choice. a. the vertical asymptote is (type an equation.) b. there is no vertical asymptote.

Explanation:

Step1: Find the vertical asymptote

For a rational function \(f(x)=\frac{N(x)}{D(x)}\), the vertical asymptote occurs at the value of \(x\) that makes \(D(x) = 0\). Here, \(D(x)=x - 2\). Set \(x-2=0\), then \(x = 2\).

Step2: Analyze the degrees of numerator and denominator

The degree of the numerator \(N(x)=5x + 5\) (degree \(n = 1\)) and the degree of the denominator \(D(x)=x-2\) (degree \(m = 1\)). When \(n=m\), the horizontal asymptote is \(y=\frac{a_n}{b_m}\), where \(a_n\) is the leading coefficient of \(N(x)\) and \(b_m\) is the leading coefficient of \(D(x)\). Here, \(a_n = 5\) and \(b_m=1\), so \(y = 5\). There is no oblique asymptote because the degree of the numerator is equal to the degree of the denominator.

Answer:

Vertical asymptote: \(x = 2\), Horizontal asymptote: \(y=5\)