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LIMITS AND DERIVATIVES — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLIMITS AND DERIVATIVES4 Marks
Question
Find $\lim\limits_{\text{x}\rightarrow0}\text{f}(\text{x})$, where $\text{f}(\text{x})\begin{cases}\frac{\text{x}}{|\text{x}|}, &\text{x}\neq 0\\0, & \text{x} = 0\end{cases}$
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Answer

The given function is $\text{f}(\text{x})\begin{cases}\frac{\text{x}}{|\text{x}|}, &\text{x}\neq 0\\0, & \text{x} = 0\end{cases}$ $\lim\limits_{\text{x}\rightarrow0^-}\text{f}(\text{x})=\lim\limits_{\text{x}\rightarrow0^-}\big[\frac{\text{x}}{|\text{x}|}\big]$ $=\lim\limits_{\text{x}\rightarrow0}\big(\frac{\text{x}}{\text{-x}}\big)$ [When x<0,$|\text{x}|=-\text{x}$] $=\lim\limits_{\text{x}\rightarrow0}(-1)$ $=-1$$\lim\limits_{\text{x}\rightarrow0^+}\text{f}(\text{x})=\lim\limits_{\text{x}\rightarrow0^+}\big[\frac{\text{x}}{|\text{x}|}\big]$
$=\lim\limits_{\text{x}\rightarrow0}\big[\frac{\text{x}}{\text{x}}\big]$ [When x>0,$|\text{x}|=\text{x}$] $=\lim\limits_{\text{x}\rightarrow0}(1)$ $=1$ It is observed that$\lim\limits_{\text{x}\rightarrow0^-}\text{f}(\text{x})\neq\lim\limits_{\text{x}\rightarrow0^+}\text{f}(\text{x})$ . Hence, $\lim\limits_{\text{x}\rightarrow0}\text{f}(\text{x})$ does not exist.

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