Find the domain and the vertical and horizontal asymptotes. Separate your asymptote answers with commas, if there are multiple, or enter "DNE" if there are none. \( f(x)=\frac{5 x^{2}}{x+5} \) Domain: Vertical asymptote(s): \( x= \) Horizontal asymptote(s): \( y= \)

To find the domain, vertical asymptotes, and horizontal asymptotes of the function \( f(x) = \frac{5x^2}{x+5} \), we will analyze the function step by step. ### Domain: The domain of a rational function is all real numbers except where the denominator is zero. Set the denominator equal to zero to find the values that are not in the domain: \[ x + 5 = 0 \implies x = -5 \] Thus, the domain of \( f(x) \) is all real numbers except \( x = -5 \): \[ \text{Domain: } (-\infty, -5) \cup (-5, \infty) \] ### Vertical Asymptotes: Vertical asymptotes occur where the denominator is zero and the numerator is not zero at that point. Since we found that the denominator is zero at \( x = -5 \) and the numerator \( 5x^2 \) is not zero at \( x = -5 \) (it equals \( 5(-5)^2 = 125 \)), there is a vertical asymptote at: \[ \text{Vertical asymptote(s): } x = -5 \] ### Horizontal Asymptotes: To find horizontal asymptotes, we look at the degrees of the polynomial in the numerator and the denominator. The degree of the numerator \( 5x^2 \) is 2, and the degree of the denominator \( x + 5 \) is 1. Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is no horizontal asymptote. Instead, the function will approach infinity as \( x \) approaches infinity or negative infinity. Thus, we have: \[ \text{Horizontal asymptote(s): } \text{DNE} \] ### Summary: - Domain: \( (-\infty, -5) \cup (-5, \infty) \) - Vertical asymptote(s): \( x = -5 \) - Horizontal asymptote(s): \( y = \text{DNE} \)