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If both \( \operatorname{Lim}_{x \rightarrow c^{-}} f(x) \) and \( \operatorname{Lim}_{x \rightarrow c^{+}} f(x) \) exist finitely and are equal, then the function \( f \) is said to have removable discontinuity at \( x=c \)
\( \mathrm{P} \) If both the limits i.e. \( \operatorname{Lim}_{x \rightarrow c^{-}} f(x) \) and \( \operatorname{Lim}_{x \rightarrow c^{+}} f(x) \) exist finitely and are not equal, then the function \( f \) is said to W have non-removable discontinuity at \( x=c \) and in this case \( \left|\operatorname{Lim}_{x \rightarrow c^{+}} f(x)-\operatorname{Lim}_{x \rightarrow c^{-}} f(x)\right| \) is called jump of the discontinuity.
Which of the following function has non-removable discontinuity at the origin?
(A) \( f(x)=\frac{1}{\ln |x|} \)
(B) \( f(x)=x \sin \frac{\pi}{x} \)
(C) \( f(x)=\frac{1}{1+2^{\cot x}} \)
(D) \( f(x)=\cos \left(\frac{|\sin x|}{x}\right) \)
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