Mathematics

Problem

Let

g(x)=\frac{5x^{3}-x\sqrt{x^{2}+1}}{2x^{3}+\sqrt{x^{4}+4}}.

(a) Re-express both the numerator and the denominator in a form that makes their comparative growth rates as

x\to\infty

transparent. Show each algebraic manipulation. (b) Using the expression obtained in part (a), evaluate

\displaystyle\lim_{x\to\infty}g(x)

. (c) Provide a brief explanation of why the presence of the radical terms does not alter the limit you obtained, referencing the relative magnitudes of the terms involved. a. Part (a) — Algebraic reformulation b. Part (b) — Evaluation of the limit c. Part (c) — Explanation of the role of radical terms

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