Which of the following function(s) not define | vedclass

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(D) function $f(x)$ has a removable discontinuity at $x = a$ if $\lim_{x 	o a} f(x)$ exists but is not equal to $f(a)$ (or $f(a)$ is undefined).For o

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All of the above

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(D) function $f(x)$ has a removable discontinuity at $x = a$ if $\lim_{x 	o a} f(x)$ exists but is not equal to $f(a)$ (or $f(a)$ is undefined).For option $A$: $\lim_{x 	o 0} \frac{1}{\ln |x|} = \frac{1}{-\infty} = 0$. Since the limit exists,it has a removable discontinuity.For option $B$: $\lim_{x 	o 0^+} \cos \left( \frac{\sin x}{x} ight) = \cos(1)$ and $\lim_{x 	o 0^-} \cos \left( \frac{-\sin x}{x} ight) = \cos(-1) = \cos(1)$. Since the limit exists,it has a removable discontinuity.For option $C$: $\lim_{x 	o 0} x \sin \left( \frac{\pi}{x} ight)$. Since $|\sin(\pi/x)| \le 1$,by the Squeeze Theorem,$\lim_{x 	o 0} x \sin \left( \frac{\pi}{x} ight) = 0$. Since the limit exists,it has a removable discontinuity.Therefore,all the functions have a removable discontinuity at $x = 0$.

Which of the following function$(s)$ not defined at $x = 0$ has/have a removable discontinuity at $x = 0$?

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