Question
The function $$f\left( x \right) = \sin \left( {{{\log }_e}\left| x \right|} \right),\,x \ne 0,$$ and $$1$$ is $$x = 0$$
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 $$x = 0$$
Answer :
has oscillating discontinuity at $$x = 0$$
Solution :
We have $$\mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right)$$
$$ = \mathop {\lim }\limits_{h \to 0} \sin \left( {{{\log }_e}\left| { - h} \right|} \right)$$
$$ = \mathop {\lim }\limits_{h \to 0} \sin \left( {{{\log }_e}h} \right)$$ which does not but lies between $$-1$$ and $$1$$
Similarly, $$\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right)$$ lies between $$-1$$ and $$1$$ but cannot be determined.