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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits3 Marks
Question
Prove that $\lim\limits_{\text{x}\rightarrow\text{a}^+}\ [\text{x}]=[\text{a}]$ for all $\text{a }\in\text{ R}.$ also prove that $\lim\limits_{\text{x}\rightarrow1^-}\ [\text{x}]=0.$
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Answer

$\lim\limits_{\text{x}\rightarrow\text{a}^+}\ [\text{x}]$
$\Rightarrow\lim\limits_{\text{h}\rightarrow0}[\text{a}+\text{h}]=[\text{a}]$
$\Rightarrow\lim\limits_{\text{h}\rightarrow0}\ [\text{x}]=[\text{a}]\forall\text{a }\in\text{ R}$
Also,
$\lim\limits_{\text{x}\rightarrow1^-}\ [\text{x}]$
$=\lim\limits_{\text{h}\rightarrow0}\ [1-\text{h}]$
$=0$
$\Rightarrow\lim\limits_{\text{x}\rightarrow1^-}\ [\text{x}]=0$

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