Problem
Determine the vertical asymptote(s) of the function. If none exists, state that fact. \[ f(x)=\frac{6 x-5}{x-5} \] Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has two vertical asymptotes. The leftmost asymptote is and the rightmost asymptote is (Type equations.) B. The function has three vertical asymptotes. The leftmost asymptote is , the middle asymptote is , and the rightmost asymptote is (Type equations.) C. The function has one vertical asymptote, (Type an equation.) D. The function has no vertical asymptotes.
Solution
Step 1 ：A vertical asymptote of a function is a vertical line \(x = a\) where the function approaches infinity (or negative infinity) as \(x\) approaches \(a\). In a rational function, vertical asymptotes occur at the values of \(x\) that make the denominator zero, as long as the numerator is not also zero at that value.
Step 2 ：In this case, the denominator of the function is \(x - 5\), so the vertical asymptote occurs at \(x = 5\).
Step 3 ：However, the numerator is also zero at \(x = 5\), so there is no vertical asymptote.
Step 4 ：\(\boxed{\text{The function has no vertical asymptotes.}}\)
