what are the vertical and horizontal asymptotes for the function $f(x)=\frac{3x^{2}}{x^{2}-4}$?\nhorizontal…

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

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

Answer

Explanation:

Step1: Find vertical asymptotes

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

Step2: Find horizontal asymptotes

Since the degree of the numerator $n = 2$ and the degree of the denominator $m=2$ are the same. The horizontal - asymptote is given by $y=\frac{a_{n}}{b_{m}}$, where $a_{n}$ is the leading coefficient of the numerator and $b_{m}$ is the leading coefficient of the denominator. Here, $a_{n}=3$ and $b_{m}=1$, so $y = 3$.

Answer:

D. horizontal asymptote: $y = 3$; vertical asymptote: $x=-2,x = 2$