 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/19/a/solution

C > 0

and

δ > 0

such that

∣ f (x) ∣ \leq C ∣ x ∣^{2} whenever 0 < ∣ x ∣ < δ

within the domain of

f

. Thus

f (x) / x^{2}

stays bounded in a punctured neighbourhood of zero. It need not have a limit:

f (x) = x^{2} sin (1/ x)

is one example. The statement imposes no condition on a separately assigned value

f (0)

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