Prove that lim as x goes to 0 of f(x) = 0 or lim as x goes to 0 of f(x)/x = 0

Suppose that f is a function that has the property that there is some M > 0 such that abs(f(x) <= M * abs(x)^2 for all x (is an element of) all real numbers.

|f(x) - 0| = |f(x)| =< M x^2. Since x^2 --> 0 as x --> 0 it follows that f(x) --> 0.

Furthermore, |f(x)/x| =< M|x| for non-zero x. Therefore lim_(x --> 0) |f(x)/x|

=< lim_(x --> 0) M|x| = 0; equivalently, lim_(x --> 0) f(x)/x = 0