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Limits — MATHS STD 11 Science — Question

Gujarat 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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