Let . (i) Show that and for every real . Hence obtain a numerical upper bound for .…

MATHEMATICS · Subjective · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

g(x)=\dfrac{\left(\sin x+2\sin 2x\right)\left(\cos x+3\cos 3x\right)}{x^{3}}

. (i) Show that

\left|\sin x+2\sin 2x\right|\le 3

and

\left|\cos x+3\cos 3x\right|\le 4

for every real

x

. Hence obtain a numerical upper bound for

\left|(\sin x+2\sin 2x)(\cos x+3\cos 3x)\right|

. (ii) Using the bound from part (i), prove that

\displaystyle \lim_{x \to \infty} g(x)=0

. (iii) Define

h(x)=\dfrac{\left(\sin x+2\sin 2x\right)\left(\cos x+3\cos 3x\right)}{x}

. State, with reason, whether

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

exists.