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Question Number 118140 by bemath last updated on 15/Oct/20

lim_(x→0)  (([ x^2  ])/(2x)) =?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left[\:{x}^{\mathrm{2}} \:\right]}{\mathrm{2}{x}}\:=?\: \\ $$

Answered by TANMAY PANACEA last updated on 15/Oct/20

let x=0.9  x^2 =0.81  [0.81]=0  so (([x^2 ])/(2x))=(0/(2×0.9))=0  let x=0.01  x^2 =0.0001  [x^2 ]=0  (([x^2 ])/(2×x))=(0/(2×0.01))=0  let x=−0.01   x^2 =0.0001  (([x^2 ])/(2x))=(0/(−2×0.01))=0  so  lim_(x→0)   (([x^2 ])/(2x))=0

$${let}\:{x}=\mathrm{0}.\mathrm{9}\:\:{x}^{\mathrm{2}} =\mathrm{0}.\mathrm{81}\:\:\left[\mathrm{0}.\mathrm{81}\right]=\mathrm{0} \\ $$$${so}\:\frac{\left[{x}^{\mathrm{2}} \right]}{\mathrm{2}{x}}=\frac{\mathrm{0}}{\mathrm{2}×\mathrm{0}.\mathrm{9}}=\mathrm{0} \\ $$$${let}\:{x}=\mathrm{0}.\mathrm{01}\:\:{x}^{\mathrm{2}} =\mathrm{0}.\mathrm{0001}\:\:\left[{x}^{\mathrm{2}} \right]=\mathrm{0} \\ $$$$\frac{\left[{x}^{\mathrm{2}} \right]}{\mathrm{2}×{x}}=\frac{\mathrm{0}}{\mathrm{2}×\mathrm{0}.\mathrm{01}}=\mathrm{0} \\ $$$${let}\:{x}=−\mathrm{0}.\mathrm{01}\:\:\:{x}^{\mathrm{2}} =\mathrm{0}.\mathrm{0001} \\ $$$$\frac{\left[{x}^{\mathrm{2}} \right]}{\mathrm{2}{x}}=\frac{\mathrm{0}}{−\mathrm{2}×\mathrm{0}.\mathrm{01}}=\mathrm{0} \\ $$$$\boldsymbol{{so}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\left[\boldsymbol{{x}}^{\mathrm{2}} \right]}{\mathrm{2}\boldsymbol{{x}}}=\mathrm{0} \\ $$

Commented by bemath last updated on 15/Oct/20

thank you sir

$${thank}\:{you}\:{sir} \\ $$

Commented by TANMAY PANACEA last updated on 15/Oct/20

most welcome sir

$${most}\:{welcome}\:{sir} \\ $$

Answered by Bird last updated on 16/Oct/20

[x^2 ]≤x^2 ≤[x^2 ] +1 ⇒for x>0  (([x^2 ])/(2x))≤(x^2 /(2x))≤(([x^2 ]+1)/(2x)) ⇒(([x^2 ])/(2x))≤(x/2)  and lim_(x→0) (x/2)=0 ⇒lim_(x→0^+ )   (([x^2 ])/(2x))=0

$$\left[{x}^{\mathrm{2}} \right]\leqslant{x}^{\mathrm{2}} \leqslant\left[{x}^{\mathrm{2}} \right]\:+\mathrm{1}\:\Rightarrow{for}\:{x}>\mathrm{0} \\ $$$$\frac{\left[{x}^{\mathrm{2}} \right]}{\mathrm{2}{x}}\leqslant\frac{{x}^{\mathrm{2}} }{\mathrm{2}{x}}\leqslant\frac{\left[{x}^{\mathrm{2}} \right]+\mathrm{1}}{\mathrm{2}{x}}\:\Rightarrow\frac{\left[{x}^{\mathrm{2}} \right]}{\mathrm{2}{x}}\leqslant\frac{{x}}{\mathrm{2}} \\ $$$${and}\:{lim}_{{x}\rightarrow\mathrm{0}} \frac{{x}}{\mathrm{2}}=\mathrm{0}\:\Rightarrow{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\frac{\left[{x}^{\mathrm{2}} \right]}{\mathrm{2}{x}}=\mathrm{0} \\ $$$$ \\ $$

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