MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions
Problem
Rational functions of the form R(x)=Q(x)P(x), where P and Q are non-zero polynomials with degrees n and m respectively, often exhibit predictable end behavior.
Answer the following questions about how the relative degrees n and m influence the existence and type of non-vertical asymptotes of y=R(x).
(a) Explain, without performing any specific limit calculations, why R must have a horizontal asymptote when n<m. Your explanation should reference the dominant terms of P and Q.
(b) Describe what occurs when n=m, and justify the form of any asymptote that results in this case.
(c) Provide a reasoned argument for why an oblique (slant) asymptote can occur only when n=m+1. Include in your discussion an outline of the algebraic steps that reveal the asymptote, without fully executing those steps.
(d) For each of the three degree relationships discussed above—n<m, n=m, and n=m+1—construct a specific example of a rational function R(x) that satisfies the degree condition and explicitly state the equation of its corresponding horizontal or oblique asymptote. Briefly verify that each asymptote you list is correct by citing appropriate limit behavior.