Find the vertical asymptotes, if any, of the graph of the rational function. * \( g(x)=\frac{x}{x^{2}-1} \)

To find the vertical asymptotes of the function \( g(x)=\frac{x}{x^{2}-1} \), we first identify where the denominator equals zero. Setting the denominator \( x^{2}-1 \) to zero gives us \( x^{2}-1=0 \), which factors into \( (x-1)(x+1)=0 \). This means the function has vertical asymptotes at \( x = 1 \) and \( x = -1 \) since these points make the denominator zero while not affecting the numerator. Now, let's explore the behavior of the function around these asymptotes. As \( x \) approaches 1 from the left, \( g(x) \) tends to \( -\infty \), and as it approaches from the right, it tends to \( \infty \). Similarly, as \( x \) approaches -1 from the left, \( g(x) \) heads towards \( \infty \), and from the right, it dives to \( -\infty \). This fascinating oscillation demonstrates the classic behavior of rational functions around their vertical asymptotes.