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Limits — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSLimits2 Marks
Question
Show that $\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}}{|\text{x}|}$ does not exist.
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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}}{|\text{x}|}$
We know that $|\text{x}|=\begin{cases}\text{x}, &\text{if }\text{x }\ge0\\-\text{x}, &\text{if } \text{x} < 0\end{cases}$
$\therefore\ \lim\limits_{\text{x}\rightarrow0^+}\frac{\text{x}}{|\text{x}|}=\lim\limits_{\text{x}\rightarrow0^+}\frac{\text{x}}{\text{x}}=\lim\limits_{\text{x}\rightarrow0^+}1=1$
Also, $\lim\limits_{\text{x}\rightarrow\sigma}\frac{\text{x}}{|\text{x}|}=\lim\limits_{\text{x}\rightarrow\sigma}\frac{\text{x}}{-\text{x}}=\lim\limits_{\text{x}\rightarrow\sigma}-1=-1$
$\Rightarrow\text{L.H.L}\text{ of f(x)}\ne\text{R.H.L of f(x)}$
$\Rightarrow\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}}{|\text{x}|}$ does not exist.

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