Follow the steps for graphing a rational function to graph the function \( P(x)=\frac{x^{4}+x^{2}+10}{x^{2}-100} \). I ne runction nas inree vericar asymptotes. I ne reimost asymptote is \( \qquad \) the miaale asymptote is and the rightmost asymptote is \( \square \) (Type equations. Use integers or fractions for any numbers in the equations.) D. The function has no vertical asymptole. Determine the behavior of the graph on either side of any vertical asymptates, if one exists. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. It approaches \( \infty \) on one side of the asymptote(s) at \( x=\square \) and \( -\infty \) on the other. It approaches either \( \infty \) or \( -\infty \) on both sides of the asymptote(s) at \( x= \) \( \square \) (Type integers or simplified fractions. Use a comma to separate answers as needed. Type each answer only once.) B. It approaches \( \infty \) on one side of the asymptote(s) at \( x= \) \( \square \) and \( -\infty \) on the other. (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) C. It approaches either \( \infty \) or \( -\infty \) on both sides of the asymptote(s) at \( x= \) \( \square \) (Type an integer or a simplified fraction. Use a comma to separate answers as needed. Type each answer only once.) D. The function has no vertical asymptote.

To graph the function \( P(x)=\frac{x^{4}+x^{2}+10}{x^{2}-100} \), we first note that the denominator \( x^{2} - 100 = 0 \) gives the vertical asymptotes. Solving for \( x \), we get \( x = -10, 10 \). The behavior of the graph near these asymptotes shows that as \( x \) approaches 10 from the left, \( P(x) \to -\infty \) and approaches \( \infty \) from the right. Similarly, as \( x \) approaches -10 from the left, \( P(x) \to \infty \) and from the right, \( P(x) \to -\infty \). For the horizontal asymptote, since the degrees of the polynomial in the numerator and denominator are both even and the leading term in the numerator dominates, we see that the horizontal asymptote is given by the ratio of leading coefficients, which is \( y=1 \). The vertical asymptotes are thus \( x = -10 \) and \( x = 10 \). The leftmost asymptote is \( -10 \), the middle is (not applicable since only two vertical asymptotes exist), and the rightmost asymptote is \( 10 \). As for the behavior around the vertical asymptotes, we see: - At \( x = -10 \): Approaches \( \infty \) from the left and \( -\infty \) from the right. - At \( x = 10 \): Approaches \( -\infty \) from the left and \( \infty \) from the right. So, we can fill the boxes as follows: - First box for vertical asymptotes: \( -10, 10 \) - For the behavior section: A. It approaches \( \infty \) on one side of the asymptote(s) at \( x=-10 \) and \( -\infty \) on the other, it approaches either \( -\infty \) or \( \infty \) on both sides of the asymptote(s) at \( x=10 \).