Limits Continuity And Differentiability Question 38
Question: The function $ f(x)=\sin ({\log _{e}}| x |),x\ne 0 $ , and 1 if $ x=0 $
Options:
A) Is continuous at $ x=0 $
B) Has removable discontinuity at $ x=0 $
C) Has jump discontinuity at $ x=0 $
D) Has oscillating discontinuity at $ x=0 $
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Answer:
Correct Answer: D
Solution:
We have $ \underset{x\to {0^{-}}}{\mathop{\lim }}f(x)=\underset{h\to 0}{\mathop{\lim }}\sin ({\log _{e}}| -h |) $
$ =\underset{h\to 0}{\mathop{\lim }}\sin ({\log _{e}}h) $ Which does not exist but lies between -1 and 1.
Similarly, $ \underset{x\to {0^{+}}}{\mathop{\lim }}f(x) $ lies between -1 and 1 but cannot be determined.
BETA
