Mathematics

Problem

Let

h(x)=x\,e^{-x}\sin x

for

x>0

. (i) Obtain a numerical bound for

|h(x)|

that depends only on

x

and does not involve any trigonometric function. (ii) Hence or otherwise, determine

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

. Provide a complete proof that the limit you obtain is correct.