Answer:
The vertical asymptotes are x=3 and x=-3. The horizontal asymptote is y=20.
Step-by-step explanation:
The given function is
[tex]f\left(x\right)=\frac{\left(4x+1\right)\left(5x-1\right)}{x^2-9}[/tex]
Equate the denominator equal to 0, to find the vertical asymptotes.
[tex]x^2-9=0[/tex]
Add 9 on both sides.
[tex]x^2=9[/tex]
Taking square root on both sides.
[tex]x=\pm \sqrt{9}[/tex]
[tex]x=\pm 3[/tex]
Therefore the vertical asymptotes are x=3 and x=-3.
The given function can be written as
[tex]f\left(x\right)=\frac{20 x^2 + x - 1}{x^2-9}[/tex]
Degree of numerator and denominator are same.If degree of numerator and denominator are same, then horizontal asymptote is
[tex]y=\frac{\text{Leading coefficient of numerator}}{\text{Leading coefficient of denominator}}[/tex]
[tex]y=\frac{20}{1}[/tex]
[tex]y=20[/tex]
Therefore the horizontal asymptote is y=20.
