If f(x) = l x, ; ; if , x , ,is ,rational , - x, ; if , , x , , i… - Vidyadip

Question

If $f(x) = \left\{ \begin{array}{l}x,\;\;{\rm{if\, }}x\,{\rm{ \,is \,rational\, }}\\ - x,\;{\rm{if \,\,}}x\,{\rm{\, is\, irrational\,}}\end{array} \right.,$ then $\mathop {\lim }\limits_{x \to 0} f(x)$ is

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Answer

$\mathop {\lim }\limits_{x \to {0^ - }} f(x) = \mathop {\lim }\limits_{h \to 0} f(0 - h) = \mathop {\lim }\limits_{h \to 0} f(0 - h) = 0$
and $\mathop {\lim }\limits_{x \to {0^ + }} f(x) = \mathop {\lim }\limits_{h \to 0} f(0 + h) = \mathop {\lim }\limits_{h \to 0} - (0 + h) = 0$
$\therefore \mathop {\lim }\limits_{x \to 0} \,\,f(x) = 0 , \left( {\because \mathop {\lim }\limits_{x \to {0^ - }} f(x) = \mathop {\lim }\limits_{x \to {0^ + }} f(x)} \right)$ .

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