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

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

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

Answer

Explanation:

Step1: Find Vertical Asymptotes

To find vertical asymptotes, set the denominator equal to zero and solve for ( x ). The denominator is ( x^2 - 4 ), so: [ x^2 - 4 = 0 ] Factor the difference of squares: ( (x - 2)(x + 2) = 0 ). Solving gives ( x = 2 ) or ( x = -2 ).

Step2: Find Horizontal Asymptotes

For rational functions ( \frac{f(x)}{g(x)} ) where the degrees of ( f(x) ) and ( g(x) ) are equal, the horizontal asymptote is the ratio of the leading coefficients. Here, ( f(x) = 3x^2 ) (leading coefficient 3) and ( g(x) = x^2 - 4 ) (leading coefficient 1). So the horizontal asymptote is: [ y = \frac{3}{1} = 3 ]

Answer:

horizontal asymptote: ( y = 3 ) vertical asymptote: ( x = -2 ), ( x = 2 ) (the last option)