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Question Number 160658 by HongKing last updated on 04/Dec/21

$$\mathbf{Find}:\underset{\mathbf{x}\to 0}{\lim }4\mathbf{x}\left[\frac{1}{4\mathbf{x}}\right]=?$$$$$$

Commented by mr W last updated on 04/Dec/21

$$\underset{x\to 0}{\lim 4}x\left[\frac{1}{4x}\right]$$$$=\underset{t\to \mathrm{\infty }}{\lim }\frac{\left[t\right]}{t}=L$$$$t-1<\left[t\right]⩽t$$$$\underset{t\to \mathrm{\infty }}{\lim }\frac{t-1}{t}<L⩽\underset{t\to \mathrm{\infty }}{\lim }\frac{t}{t}$$$$1<L⩽1$$$$\Rightarrow L=1$$

Commented by mnjuly1970 last updated on 04/Dec/21

$$verynicesolutionsirW$$

Commented by HongKing last updated on 04/Dec/21

$$\mathrm{Nice}\mathrm{solution}\mathrm{my}\mathrm{dear}\mathbf{Sir}\mathrm{thank}\mathrm{you}$$

Answered by mathmax by abdo last updated on 04/Dec/21

$$\mathrm{we}\mathrm{have}\left[\frac{1}{4\mathrm{x}}\right]⩽\frac{1}{4\mathrm{x}}<\left[\frac{1}{4\mathrm{x}}\right]+1\Rightarrow \frac{1}{4\mathrm{x}}-1<\left[\frac{1}{4\mathrm{x}}\right]⩽\frac{1}{4\mathrm{x}}\Rightarrow$$$$\mathrm{for}\mathrm{x}>0\mathrm{we}\mathrm{have}1-4\mathrm{x}<4\mathrm{x}.\left[\frac{1}{4\mathrm{x}}\right]⩽1\Rightarrow$$$${\lim }_{\mathrm{x}\to 0^{+}}4\mathrm{x}\left[\frac{1}{4\mathrm{x}}\right]=1$$

Commented by HongKing last updated on 04/Dec/21

$$Thankyousomuchmydear\mathit{S}\mathit{i}\mathit{r}vool$$

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