Let and , where is measured in real numbers. Give a complete analytical treatment of…

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

f(x)=e^{2x}+x

and

g(x)=e^{2x}-x^{2}

, where

x

is measured in real numbers. Give a complete analytical treatment of the end–behavior of the ratio

\displaystyle \frac{f(x)}{g(x)}

x\to \infty

. (a) Evaluate

\displaystyle \lim_{x\to\infty}\frac{e^{x}}{x^{3}}

and briefly explain the significance of your result for comparing exponential and polynomial growth. (b) Using clear calculus reasoning, determine

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

. Show all intermediate steps and justify the validity of each transition in your work. (c) State every horizontal asymptote of the graph of

y=\dfrac{f(x)}{g(x)}

, if any exist, and interpret the meaning of your answer to part (b) in that context.