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\(\lim\limits_{x \to 4}\frac{|x−4|}{x-4} \) does not exist.

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Let x − 4 = n 

⟹ As x ⟶ 4, 

n ⟶ 0

∴ L.H.S 

\(\lim\limits_{n \to 0-}\frac{|n|}{n}\)

\(\lim\limits_{n \to 0-}\frac{-n}{n}\)

\(\lim\limits_{n \to 0-}\)(-1)

= -1

R.H.S,

\(\lim\limits_{n \to 0+}\frac{|n|}{n}\)

\(\lim\limits_{n \to 0+}\frac{-n}{n}\)

\(\frac{n}{n}=\lim\limits_{n \to 0+}(1)\)

=1

Since L.H.S ≠ R.H.S

\(\lim\limits_{x \to 4}\frac{|x-4|}{x-4}\) does not exist

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