Find the vertical asymptotes, if any, and the values of x corresponding to holes , if any, of the graph of th f(x)=(x)/(x^2)+3 Select the correct choice below and, if necessary , fill in the answer box to complete your choice. (Type an A. The vertical asymptote(s) is (are) x= and hole(s)corresponding to x= B. The vertical asymptote(s) is (are) x= There are no holes. C. There are no vertical asymptotes but there is (are) hole(s)corresponding to x= D. There are no discontinuities.

To determine the vertical asymptotes and holes of the function \( f(x) = \frac{x}{x^2 + 3} \), we need to analyze the denominator, as vertical asymptotes occur where the denominator is zero (and the numerator is not zero at those points).<br /><br />1. **Finding Vertical Asymptotes:**<br /> The denominator of the function is \( x^2 + 3 \). We set it equal to zero to find the values of \( x \) that make the denominator zero:<br /> \[<br /> x^2 + 3 = 0<br /> \]<br /> Solving for \( x \):<br /> \[<br /> x^2 = -3<br /> \]<br /> Since \( x^2 = -3 \) has no real solutions (as the square of a real number cannot be negative), there are no values of \( x \) that make the denominator zero. Therefore, there are no vertical asymptotes.<br /><br />2. **Finding Holes:**<br /> Holes in the graph occur when both the numerator and the denominator have common factors that can be canceled out. In this case, the numerator is \( x \) and the denominator is \( x^2 + 3 \). There are no common factors between the numerator and the denominator, so there are no holes in the graph.<br /><br />Based on this analysis, the correct choice is:<br /><br />D. There are no discontinuities.