2. A rational function has the form $f(x)=\frac{a x^{2}}{x^{n}-5}$ 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. (2 marks)
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. ( 2 marks)

Step 1 ：A vertical asymptote occurs when the denominator of a rational function is equal to zero. In this case, the denominator is \(x^{n}-5\). This will be zero when \(x^{n}=5\). Since \(n\) is a positive integer, there will always be a real value of \(x\) that makes the denominator zero, regardless of the value of \(n\). Therefore, the function will always have a vertical asymptote, and there are no values of \(a\) and \(n\) for which the function will have no vertical asymptotes.

Step 2 ：A horizontal asymptote occurs when the degree of the numerator is less than or equal to the degree of the denominator. In this case, the degree of the numerator is 2 and the degree of the denominator is \(n\). Therefore, the function will have a horizontal asymptote if \(n \geq 2\).

Step 3 ：Final Answer: \(\boxed{\text{a) No, there are no values of } a \text{ and } n \text{ for which the function will have no vertical asymptotes. }}\)

Step 4 ：Final Answer: \(\boxed{\text{b) Yes, the function will have a horizontal asymptote if } n \geq 2}\)