Question

find the vertical asymptote, if any, and the values of x corresponding to holes, if any, of the graph of the rational function.
h(x) = \frac{x + 3}{x(x - 6)}

select the correct choice below and, if necessary, fill in the answer box to complete your choice. (type an equation. use a comma to separate answers as needed.)
\bigcirc a. the vertical asymptote(s) is(are) \square and hole(s) corresponding to \square.
\bigcirc b. the vertical asymptote(s) is(are) \square. there are no holes.
\bigcirc c. there are no vertical asymptotes but there is(are) hole(s) corresponding to \square.
\bigcirc d. there are no discontinuities.

Explanation:

Step1: Factor numerator and denominator

The function is \( h(x) = \frac{x + 3}{x(x - 6)} \). The numerator \( x + 3 \) can't be factored further, and the denominator is already factored as \( x(x - 6) \).

Step2: Check for common factors

There are no common factors between the numerator \( x + 3 \) and the denominator \( x(x - 6) \).

Step3: Find vertical asymptotes

Vertical asymptotes occur where the denominator is zero (and numerator is not zero). Set denominator \( x(x - 6)=0 \). Solving \( x = 0 \) or \( x - 6 = 0 \) (i.e., \( x = 6 \)). Since numerator is not zero at \( x = 0 \) or \( x = 6 \), these are vertical asymptotes.

Step4: Check for holes

Holes occur where numerator and denominator have common factors. Since there are no common factors, there are no holes.

Answer:

B. The vertical asymptote(s) is(are) \( x = 0, x = 6 \). There are no holes.