Answer:
Option D.
Step-by-step explanation:
The given function is
[tex]f\left(x\right)=\dfrac{3x^{2}-3}{x^{2}-4}[/tex]
In this function the degree of numerator an denominator is same i.e., 2.
Horizontal Asymptotes: If the degree of numerator an denominator is same, then
[tex]\text{Horizontal asymptote}=\frac{\text{Leading coefficient of numerator}}{\text{Leading coefficient of denominator}}[/tex]
[tex]\text{Horizontal asymptote}=\frac{3}{1}[/tex]
[tex]\text{Horizontal asymptote}=3[/tex]
Horizontal asymptote is y=3.
[tex]f(x)=3[/tex]
[tex]\dfrac{3x^{2}-3}{x^{2}-4}=3[/tex]
[tex]3(x^{2}-1)=3(x^{2}-4)[/tex]
[tex]x^{2}-1=x^{2}-4[/tex]
[tex]1=4[/tex]
This statement is false for any value of x, therefore the graph does not cross the horizontal asymtote.
Vertical Asymptotes: Equate the denominator equal to 0, to find the vertical asymptotes.
[tex]x^2-4=0[/tex]
Add 4 on both sides.
[tex]x^2=4[/tex]
Taking square root on both sides.
[tex]x=\pm \sqrt{4}[/tex]
[tex]x=\pm 2[/tex]
Vertical asymptotes are x=2 and x=-2.
The graph has two vertical asymptotes and one horizontal asymptote.
Therefore, the correct option is D.
