# Prove that the $$\lim_{x \to 2} \frac{|x – 2|}{x – 2}$$ doesn’t exist.

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Solution:

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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