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Question Number 79992 by john santu last updated on 29/Jan/20

$$\underset{x\to 0}{\lim }\left[\frac{1}{\mathrm{x}}\right]=?$$

Commented by jagoll last updated on 30/Jan/20

$$\mathrm{do}\mathrm{not}\mathrm{exis}$$

Commented by mathmax by abdo last updated on 30/Jan/20

$$x<\left[x\right]+1\Rightarrow \left[x\right]>x-1\Rightarrow \left[\frac{1}{x}\right]>\frac{1}{x}-1wehave{lim}_{x\to 0^{+}}(\frac{1}{x}-1)=+\mathrm{\infty }$$$$\Rightarrow {lim}_{x\to 0^{+}}\left[\frac{1}{x}\right]=+\mathrm{\infty }forx<0\left[\frac{1}{x}\right]⩽\frac{1}{x}wehave{lim}_{x\to 0^{-}}\frac{1}{x}=-\mathrm{\infty }$$$$\Rightarrow {lim}_{x\to 0^{-}}\left[\frac{1}{x}\right]=-\mathrm{\infty }$$

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