Functions Question 199
Question: If $ f(x)= \begin{cases} & x,\ \ \text{if }x\text{ is rational } \\ & -x,\ \text{if }x\text{ is irrational} \\ \end{cases} ., $ then $ \underset{x\to 0}{\mathop{\lim }},f(x) $ is
[Kurukshetra CEE 1998; UPSEAT 2004]
Options:
A) Equal to 0
B) Equal to 1
C) Equal to ?1
D) Indeterminate
Show Answer
Answer:
Correct Answer: A
Solution:
$ \underset{x\to {0^{-}}}{\mathop{\lim }},f(x)=\underset{h\to 0}{\mathop{\lim }},f(0-h)=\underset{h\to 0}{\mathop{\lim }},f(0-h)=0 $ and $ \underset{x\to {0^{+}}}{\mathop{\lim }},f(x)=\underset{h\to 0}{\mathop{\lim }},f(0+h)=\underset{h\to 0}{\mathop{\lim }}-(0+h)=0 $
$ \therefore ,\underset{x\to 0}{\mathop{\lim }},f(x)=0 $ , $ ( \because \underset{x\to {0^{-}}}{\mathop{\lim }},f(x)=\underset{x\to {0^{+}}}{\mathop{\lim }},f(x) ) $ .
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