Mathematics

Problem

Let

f(x)=\dfrac{\sin^2(2x)}{x}

, defined for all real

x\ge 1

. (a) Show that

0\le \sin^2(2x)\le 1

for all real

x

. Provide a brief justification. (b) Use the inequality from part (a) to obtain numerical upper and lower bounds for

f(x)

that are simpler rational functions of

x

. (c) Using the bounds from part (b), determine

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

and justify your result rigorously.