Question

find the vertical asymptotes, 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.)
a. the vertical asymptote(s) is(are) \\( \square \\) and hole(s) corresponding to \\( \square \\).
b. the vertical asymptote(s) is(are) \\( \square \\). there are no holes.
c. there are no vertical asymptotes but there is(are) hole(s) corresponding to \\( \square \\).
d. there are no discontinuities.

Explanation:

Step1: Identify Denominator Zeros

To find vertical asymptotes and holes, first find where the denominator is zero. The function is \( h(x) = \frac{x + 3}{x(x - 6)} \). The denominator \( x(x - 6) = 0 \) when \( x = 0 \) or \( x = 6 \).

Step2: Check Numerator for Common Factors

The numerator is \( x + 3 \), which has no common factors with the denominator \( x(x - 6) \). So, there are no holes (since holes require common factors in numerator and denominator), and vertical asymptotes occur where the denominator is zero (and numerator is non - zero there).

Step3: Determine Vertical Asymptotes

Since \( x = 0 \) and \( x = 6 \) make the denominator zero and the numerator non - zero at these points, the vertical asymptotes are \( x = 0 \) and \( x = 6 \), and there are no holes. So option B is correct, with vertical asymptotes \( x = 0, x = 6 \).

Answer:

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