# Advanced Polynomial & Rational Functions

1. A rational function has the form
$𝑓(𝑥) = \frac{𝑎𝑥^2}{𝑥^𝑛 − 5}$
where a is a non-zero real number and n is a positive integer.

a) Are there any values of "a" and n for which the function above will have no vertical asymptotes? Explain why by providing examples and showing work. Use terminology learned in this unit.

b) Are there any values of "a" and "n" for which the function above will have a horizontal asymptote? Explain why by providing examples and showing work. Use terminology learned in this unit.

a) For the function $𝑓(𝑥) =\frac{𝑎𝑥^2}{𝑥^𝑛-5}$ to have no vertical asymptote, you must not be deviding by zero. This means that the exponent of $x^n-5=0$ should not have a solution. However, if $n\neq 0$, then

$x=\sqrt[n]{5}$
is a solution of $x^n-5=0$. So in order to not have a veritical asymptote we must have $n=0$. Also of $a=0$, then $f(x)=0$ and it does not have a veritical asymptote. In summary we should have
$a=0 \text{or} n=0,$
to not have any vertical asymptote.

b) For this function $𝑓(𝑥) =\frac{𝑎𝑥^2}{𝑥^𝑛-5}$ to have a horizontal asymptote, the following limit should be a finite number
$\lim_{x \rightarrow \pm \infty} \frac{𝑎𝑥^2}{𝑥^𝑛-5}.$
Hence we should have $n\geq 2$. Also if  $a=0$ then $f(x)=0$, which has the veritical asymptote $y=0$. So to have a horizontal asymptote we must have
$a=0 \text{or} n\geq 2.$

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