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We have to evaluate

\(\lim_{x \to 2^{-}} \left ( \frac{|x – 2|}{x – 2} \right )\)

Here,

\(LHL = \lim_{x \to 2^{-}} \left ( \frac{|x – 2|}{x – 2} \right )\)

\(= \lim_{x \to 2^{-}} \left ( \frac{-(x – 2)}{x – 2} \right )\) = -1

and,

\(RHL = \lim_{x \to 2^{+}} \left ( \frac{|x – 2|}{x – 2} \right )\)

\(= \lim_{x \to 2^{-}} \left ( \frac{(x – 2)}{x – 2} \right )\) = 1

This shows LHL **≠ **RHL. So, the limit of given function doesn’t exists.

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