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

MCQ

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

✓
$ 0$
B
$ 1$
C
$ -1$
D
Indeterminate

✓

Answer

Correct option: A.

$ 0$

a
(a) $\mathop {\lim }\limits_{x \to {0^ - }} f(x) = \mathop {\lim }\limits_{h \to 0} f(0 - h) = \mathop {\lim }\limits_{h \to 0} f(0 - h) = 0$

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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