The value of ₓ→ 0}|x|/x is

Correct Answer is : Does not exist

Correct Answer is : Does not exist

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UPSC NDA Math Model Paper 1

Question : 71 of 120

Marks: +1, -0

The value of

\lim\limits_{x\to 0}\frac{|x|}{x}

Solution:

For a limit to exist Left-hand limit and right-hand limit must be equal.

|x| can have two values

|x | = - x when x is negative

|x| = x when x is positive.

\lim\limits_{x\to 0^{-}}\frac{|x|}{x} = \lim\limits_{x\to 0}\frac{-x}{x} = -1

\lim\limits_{x\to 0^{+}}\frac{|x|}{x} = \lim\limits_{x\to 0}\frac{x}{x} = 1

Here,

\lim\limits_{x\to 0^{-}}\frac{|x|}{x} \neq \lim\limits_{x\to 0^{+}}\frac{|x|}{x}

Hence,

\lim\limits_{x\to 0}\frac{|x|}{x}

does not exist.

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