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 zero: $x^{2}-4 = 0$. Factor the equation: $(x + 2)(x - 2)=0$. Solve for $x$: $x=-2$ or $x = 2$.

Step2: Find horizontal asymptotes

Since the degree of the numerator and denominator are the same (both degree 2), divide the leading - coefficients. The leading coefficient of the numerator is 3 and of the denominator is 1. So, $\lim_{x\rightarrow\pm\infty}\frac{3x^{2}}{x^{2}-4}=\lim_{x\rightarrow\pm\infty}\frac{3}{1-\frac{4}{x^{2}}}=3$.

Answer:

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