Question

what are the vertical and horizontal asymptotes for the function (f(x)=\frac{3x^{2}}{x^{2}-4})?
horizontal asymptote: (y = 3) vertical asymptote: (x=-2,x = 2)
horizontal asymptote: (y=-4,y = 1) vertical asymptote: (x = 3)
horizontal asymptote: (y=-2,y = 2) vertical asymptote: (x = 3)

Explanation:

Step1: Find vertical asymptotes

Set the denominator equal to 0. Given $f(x)=\frac{3x^{2}}{x^{2}-4}$, we solve $x^{2}-4 = 0$. Factoring, we get $(x + 2)(x - 2)=0$. So $x=-2$ or $x = 2$.

Step2: Find horizontal asymptotes

Since the degree of the numerator and the denominator are the same (both degree 2), we divide the leading - coefficients. The leading coefficient of the numerator is 3 and of the denominator is 1. So $y=\frac{3}{1}=3$.

Answer:

Horizontal asymptote: $y = 3$, vertical asymptote: $x=-2,x = 2$