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(D) function has an irremovable discontinuity at $x = a$ if the limit $\lim_{x 	o a} f(x)$ does not exist.$(1)$ For $f(x) = \frac{1}{\ln |x|}$,$\lim_

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$f(x) = \frac{1}{1 + 2^{\cot x}}$

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(D) function has an irremovable discontinuity at $x = a$ if the limit $\lim_{x 	o a} f(x)$ does not exist.$(1)$ For $f(x) = \frac{1}{\ln |x|}$,$\lim_{x 	o 0} \frac{1}{\ln |x|} = 0$. Since the limit exists,it has a removable discontinuity.$(2)$ For $f(x) = \cos \left( \frac{|\sin x|}{x} ight)$:$\lim_{x 	o 0^+} \cos \left( \frac{\sin x}{x} ight) = \cos(1)$$\lim_{x 	o 0^-} \cos \left( \frac{-\sin x}{x} ight) = \cos(-1) = \cos(1)$.Since the limit exists,it has a removable discontinuity.$(3)$ For $f(x) = x \sin \frac{\pi}{x}$,by the Squeeze Theorem,$\lim_{x 	o 0} x \sin \frac{\pi}{x} = 0$. Since the limit exists,it has a removable discontinuity.$(4)$ For $f(x) = \frac{1}{1 + 2^{\cot x}}$:$\lim_{x 	o 0^+} \frac{1}{1 + 2^{\cot x}} = \frac{1}{1 + 2^{\infty}} = 0$$\lim_{x 	o 0^-} \frac{1}{1 + 2^{\cot x}} = \frac{1}{1 + 2^{-\infty}} = \frac{1}{1 + 0} = 1$.Since the left-hand limit $(1)$ is not equal to the right-hand limit $(0)$,the limit does not exist. Thus,it has an irremovable (jump) discontinuity at $x = 0$.

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

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