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)equalsStartFraction x plus 3 Over x left parenthesis x plus 6 right parenthesis EndFraction
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Part 1
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)
  
enter your response here. There are no holes.
B.
There are no vertical asymptotes but there​ is(are) hole(s) corresponding to
  
enter your response here.
C.
The vertical​ asymptote(s) is(are)
  
enter your response here and​ hole(s) corresponding to
  
enter your response here.
D.
There are no discontinuities.

Answer 1

To find the vertical asymptotes and holes of the function \( h(x) = \frac{x + 3}{x(x + 6)} \), we need to examine the denominator and see where it equals zero.

1. **Finding Vertical Asymptotes**:
- The vertical asymptotes occur where the denominator is equal to zero, but the numerator is not simultaneously equal to zero.
- The denominator \( x(x + 6) \) equals zero when:
\[
x = 0 \quad \text{or} \quad x + 6 = 0 \quad \Rightarrow \quad x = -6
\]
- Thus, the vertical asymptotes are at \( x = 0 \) and \( x = -6 \).

2. **Finding Holes**:
- A hole would occur if a factor in the denominator also appears in the numerator. In this function, the numerator \( x + 3 \) does not have common factors with the denominator \( x(x + 6) \).
- Therefore, there are no holes.

Summarizing the results:
- Vertical asymptotes: \( x = 0, -6 \)
- Holes: none

The final answer matches with choice **A**:

**A. The vertical​ asymptote(s) is(are)** \( x = 0, -6 \). **There are no holes.**