Mathematics

Problem

Consider the three functions

a(x)=x\sin x,\qquad b(x)=x,\qquad c(x)=x\ln x,

all defined for

x>0

. Although

a

oscillates, its magnitude can still be compared with the growth of

b

and

c

. (a) Prove that

|\sin x|\le 1

for all real

x

and explain how this bound constrains the size of

a(x)

. (b) Determine

\displaystyle \lim_{x\to\infty}\frac{|a(x)|}{b(x)}

and

\displaystyle \lim_{x\to\infty}\frac{|a(x)|}{c(x)}

. Carefully justify any conclusion that a limit equals

0

. (c) Evaluate

\displaystyle \lim_{x\to\infty}\frac{b(x)}{c(x)}

and state the implication for the relative growth of

b

and

c

. Explain your reasoning. (d) Using the work from parts (b) and (c), rank

a

b

, and

c

in order of increasing magnitude as

x\to\infty

. Provide a concise justification referencing your limits.