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On a bounded function and its limits

The following problem appeared in an examination of Calculus II and it is quite fun.

Problem

  1. Give an example of a bounded function f:(0, +\infty) \rightarrow \mathbb{R} such that the limit \ell= \lim \limits_{x \rightarrow 0^+} f(x) does not exist. Give a brief explanation.
  2. Let f be a function of the previous question \text{(i)}. Examine whether the following limits exist: 

        \[\ell_1 = \lim_{x\rightarrow 0^+} x f(x) \quad , \quad \ell_2 = \lim_{x \rightarrow 0^+} \left ( 1- x \right ) f(x)\]

    Give a brief explanation.

Solution

  1. One such function , for example , is the following:

        \[f(x) = \left\{\begin{matrix} 1 & , & x \in \mathbb{Q} \\ 0 & , & x \in \mathbb{R} \setminus \mathbb{Q} \end{matrix}\right.\]

  2. For the \ell_1 limit we have that the limit is 0 since

        \[\left | x f(x) \right | \leq M\left | x \right |\]

    where M is the bound of |f|. Hence by squeezing the result follows. As of \ell_2 we have that the limit , in general , does not exist since we can write

        \[f(x)  = x f(x) + (1 - x) f(x)\]

    and as we have seen in the first question the limit of f as x \rightarrow 0^+ does not exist.

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