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$$ \lim_{x \to 0} \frac{a}{x} \left\lfloor\frac{x}{b} \right\rfloor $$

The $\lfloor \rfloor$ stands for the greatest integer function.

I have calculated and the left-hand limit is coming as (ab). But, I have doubt in the right-hand limit. I did this problem by sandwich-theorem. Can, anyone help me to find the right-hand limit correctly?

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1 Answer 1

Hint: Assuming $b \ne 0$,

$$ \lim_{x \to 0} \frac{a}{x} \left\lfloor\frac{x}{b} \right\rfloor = \lim_{x \to 0} \frac{a}{bx} \left\lfloor \frac{bx}{b}\right\rfloor = \lim_{x \to 0} \frac{a}{b} \left( \frac{\lfloor x\rfloor}{x}\right) = \frac{a}{b} \lim_{x \to 0} \frac{\lfloor x\rfloor}{x} $$

For $x$ near $0$, $\lfloor x \rfloor$ is just $0$ on the right and $-1$ on the left.

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