Question

find the horizontal and vertical asymptotes of f(x).
f(x)=\frac{x^{6}}{x^{2}+5}
find the horizontal asymptotes. select the correct choice below and fill in any answer boxes within your choice.
a. the horizontal asymptote(s) can be described by the line(s)
(type an equation. use a comma to separate answers as needed.)
b. there are no horizontal asymptotes.

Explanation:

Step1: Recall horizontal - asymptote rules

For a rational function $f(x)=\frac{a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_0}{b_mx^m + b_{m - 1}x^{m-1}+\cdots+b_0}$, if $n>m$, there is no horizontal asymptote. Here, $n = 6$ (degree of numerator) and $m = 2$ (degree of denominator) in $f(x)=\frac{x^6}{x^2 + 5}$.

Step2: Analyze vertical - asymptote

Set the denominator equal to zero: $x^2+5 = 0$. Then $x^2=-5$. Since there is no real - valued solution for $x$ (because for any real number $x$, $x^2\geq0$), there are no vertical asymptotes.

Answer:

B. There are no horizontal asymptotes.