For each real number

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

k

, define the functions

F_k(x)=x^k e^{-x}\quad\text{and}\quad G(x)=\ln x,

for

x>0

. The parameter

k

influences how quickly

F_k

grows or decays relative to

G

. (a) For a fixed integer

k

, determine

\displaystyle \lim_{x\to\infty}\frac{F_k(x)}{G(x)}

. Show all algebraic steps. (b) Using the result from part (a), find all real values of

k

for which

F_k

eventually grows faster than

G

. Justify your conclusion in terms of the limit from part (a). (c) For the critical value(s) of

k

at which the long-run dominance switches, compute

\displaystyle \lim_{x\to\infty}\frac{G(x)}{F_k(x)}

and interpret what this says about the comparative growth of

F_k

and

G

at that value of

k

. Include a sentence connecting your limit to a horizontal asymptote of the quotient.