Problem

Let

h(x)=x\sin\bigl(\dfrac{1}{x^{2}}\bigr)

. (a) Establish inequalities that bound

|h(x)|

for all sufficiently large

x>0

. (b) Using the bounds obtained in part (a), determine

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

. (c) Comment briefly on how the behaviour of

\dfrac{1}{x^{2}}

x\to\infty

influences the limit obtained.