find any horizontal or vertical asymptotes. f(x) = (x^4 + 1)/(x^2 + 4x - 12) select the correct choice below…

find any horizontal or vertical asymptotes. f(x) = (x^4 + 1)/(x^2 + 4x - 12) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the horizontal asymptotes are y = (use a comma to separate answers as needed.) b. there are no horizontal asymptotes.

find any horizontal or vertical asymptotes. f(x) = (x^4 + 1)/(x^2 + 4x - 12) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the horizontal asymptotes are y = (use a comma to separate answers as needed.) b. there are no horizontal asymptotes.

Answer

Explanation:

Step1: Determine horizontal asymptotes

For a rational function $f(x)=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}$, we compare degrees of numerator $n$ and denominator $m$. Here $n = 4$ and $m=2$. Since $n>m$, there are no horizontal asymptotes.

Step2: Determine vertical asymptotes

Factor the denominator $x^{2}+4x - 12=(x + 6)(x-2)$. Set the denominator equal to zero: $(x + 6)(x - 2)=0$. Solving gives $x=-6$ and $x = 2$.

Answer:

Horizontal asymptotes: B. There are no horizontal asymptotes. Vertical asymptotes: $x=-6,x = 2$