Question

Find the vertical​ asymptotes, if​ any, and the values of x corresponding to​ holes, if​ any, of the graph of the rational function.
​f(x)equalsStartFraction x Over x minus 3 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 commas 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 analyze the function \( f(x) = \frac{x}{x - 3} \) for vertical asymptotes and holes, we need to consider the denominator.

1. **Finding Vertical Asymptotes**:
Vertical asymptotes occur where the denominator is zero and the numerator is not zero at those points.

Set the denominator equal to zero:
\[
x - 3 = 0 \implies x = 3
\]
The numerator at \( x = 3 \) is:
\[
f(3) = \frac{3}{3 - 3} \quad \text{(undefined)}
\]
Therefore, there is a vertical asymptote at \( x = 3 \).

2. **Finding Holes**:
Holes occur in the graph if there are factors that cancel in the numerator and the denominator. Since there are no common factors between the numerator and denominator in this function, there are no holes.

Based on this analysis:

**Final Answer**:
The correct choice is:
A. The vertical asymptote(s) is(are) \( x = 3 \). There are no holes.