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Question: \[\lim_{x \rightarrow 2}\frac{|x - 2|}{x - 2} =\]...

lim⁡x→2∣x−2∣x−2=\lim_{x \rightarrow 2}\frac{|x - 2|}{x - 2} =

A

1

B

–1

C

Does not exist

D

None of these

Answer

Does not exist

Explanation

Solution

L.H.L.=lim⁡x→2−∣x−2∣x−2=lim⁡h→0∣2−h−2∣2−h−2\lim_{x \rightarrow 2^{-}}\frac{|x - 2|}{x - 2} = \lim_{h \rightarrow 0}\frac{|2 - h - 2|}{2 - h - 2}

= lim⁡h→0h−h=−1\lim_{h \rightarrow 0}\frac{h}{- h} = - 1…..(i)

and, R.H.L.=lim⁡x→2+∣x−2∣x−2=lim⁡h→0∣2+h−2∣2+h−2\lim_{x \rightarrow 2^{+}}\frac{|x - 2|}{x - 2} = \lim_{h \rightarrow 0}\frac{|2 + h - 2|}{2 + h - 2} = lim⁡h→0hh=1\lim_{h \rightarrow 0}\frac{h}{h} = 1…..(ii)

From (i) and (ii) L.H.L. ≠ R.H.L. i.e. lim⁡x→2∣x−2∣x−2\lim_{x \rightarrow 2}\frac{|x - 2|}{x - 2} does not exist.