Friday, 5 February 2016

real analysis - limlimitsxto0+f(x) and limlimitsxto0+g(x) do not exist, limlimitsxto0+f(x)g(x) does



I am looking for functions f,g:(0,+)(0,+) such that lim and \lim\limits_{x \to 0^+}g(x) do not exist, however, \lim\limits_{x \to 0^+}f(x)g(x) does. So far, I have only encountered two functions for which the one-sided limit doesn't exist: \sin{1 \over x} and \cos{1 \over x}. This doesn't seem to be what I'm looking for. I would appreciate any hints.


Answer



For example: f(x)=\begin{cases} 2 & \text{if}& x\text{ rational}\\1/2 & \text{if}& x\text{ irrational},\end{cases}\qquad g(x)=\begin{cases} 2 & \text{if}& x\text{ irrational}\\1/2 & \text{if}& x\text{ rational.}\end{cases}


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