Prof. Banchoff made the following comment in class on Monday. The
expression x^2 sin(3/x) can be written 3x (sin(3/x)/(3/x)). That
is correct, and it will be useful when we study what happens
when x goes to infinity. It is not useful, however, when we want
to study the limit as x goes to zero, since the primary fact we
need to use is that the limit as x goes to zero of sin(u(x))/u(x)
goes to 1 if the limit of u(x) is zero as x goes to zero. Since
3/x does not go to zero as x goes to zero, we can't use this basic
fact to compute the limit of the original expression as x goes to zero.
The method that does give the limit of x^2 sin(3/x) as x goes to zero is the squeeze principle, as explained in the solution key.