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Question Number 72806 by Rio Michael last updated on 03/Nov/19

$$\underset{x\to 0}{showthat\lim }\left[x\right]doesnotexist.$$$$Hencedefine\left[x\right]andsketchagraphfor$$$$y=3x^2+\left[x\right]$$

Commented by mathmax by abdo last updated on 03/Nov/19

$$letf\left(x\right)=\left[x\right]wehave{lim}_{x\to 0^{+}}f\left(x\right)={lim}_{x\to 0^{+}}\left[x\right]=0$$$${lim}_{x\to 0^{-}}f\left(x\right)={lim}_{x\to 0^{-}}\left[x\right]=-1andf\left(0\right)=0wehave0\ne -1\Rightarrow$$$${lim}_{x\to 0}\left[x\right]dontexist$$

Commented by Rio Michael last updated on 03/Nov/19

$$lovethatmethod,thankssir$$

Answered by mind is power last updated on 03/Nov/19

$$\mathrm{suppose}\mathrm{that}\underset{x\to 0}{\lim }\left[\mathrm{x}\right]\mathrm{exist}$$$$\mathrm{call}\mathrm{l}=\underset{x\to 0}{\lim }\left[\mathrm{x}\right]$$$$\Rightarrow \mathrm{\forall }\epsilon >0\mathrm{\exists }\eta >0\mid \mid \mathrm{x}\mid <\eta \Rightarrow \mid \left[\mathrm{x}\right]-\mathrm{l}\mid <\epsilon$$$$\mathrm{let}ϵ=\frac{\mathrm{l}}{3}$$$$\Rightarrow \mathrm{\exists }{\eta }_{\epsilon }\mathrm{such}\mathrm{that}\mathrm{\forall }\mathrm{x}\in ]-\eta ,\eta [\mid \left[\mathrm{x}\right]-\mathrm{L}\mid <\frac{1}{3}$$$$\mathrm{for}\mathrm{x}\in ]-\eta ,0[\Rightarrow \mid -1-\mathrm{L}\mid <\frac{1}{2}\iff \mid 1+\mathrm{L}\mid <\frac{1}{3}$$$$\mathrm{for}\mathrm{x}\in ]0,\eta [\Rightarrow \mid -\mathrm{L}\mid <\frac{1}{3}$$$$\mathrm{we}\mathrm{have}\mid 1+\mathrm{L}-\mathrm{L}\mid =1$$$$\mathrm{truangular}\mathrm{inquality}\Rightarrow \mid 1\mid =1⩽\mid -\mathrm{L}\mid +\mid 1+\mathrm{L}\mid ⩽\frac{1}{3}+\frac{1}{3}=\frac{2}{3}$$$$\mathrm{absurd}\mathrm{since}\mathrm{that}\mathrm{L}/\mathrm{dosent}\mathrm{exist}$$$$$$$$2)$$$$\mathrm{y}={3\mathrm{x}}^2+\mathrm{k}\mathrm{x}\in [\mathrm{k},\mathrm{k}+1[$$

Commented by Rio Michael last updated on 03/Nov/19

$$thanks$$

Commented by mind is power last updated on 03/Nov/19

$$\mathrm{most}\mathrm{Welcom}$$

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