Mathematics

Mathematics

Problem

Define the piecewise rational function h(x)={4x5−x2+62x5+9,x≥0,−5x5+3xx5−2x2+1,x<0.h(x)=\begin{cases} \dfrac{4x^{5}-x^{2}+6}{2x^{5}+9}, & x\ge 0,\\ \dfrac{-5x^{5}+3x}{x^{5}-2x^{2}+1}, & x<0. \end{cases} (a) Evaluate lim⁡x→∞h(x)\displaystyle\lim_{x\to\infty}h(x), showing complete algebraic justification. (b) Evaluate lim⁡x→−∞h(x)\displaystyle\lim_{x\to-\infty}h(x), again providing full justification. (c) Based on your work in parts (a) and (b), discuss whether hh possesses any horizontal asymptote(s). If so, state their equations and explain how each relates to the respective one-sided limit. a. Part (a) — Limit as x→∞x\to\infty b. Part (b) — Limit as x→−∞x\to-\infty c. Part (c) — Discussion of horizontal asymptote(s)

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