Prove that limit x-0 |x| / x doesn’t exists

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Prove that limit x->0 |x| / x doesn&#8217;t exists

Suresh Chand · Accepted Answer

Solution: Given, (lim_{x to 0} frac{|x|}{x}) Since we have, (|x| = begin{Bmatrix} x  enspace for enspace x geq 0 -x enspace for enspace x < 0 end{Bmatrix}) Then (frac{|x|}{x} = begin{Bmatrix} frac{+x}{x} enspace for enspace x geq 0 frac{-x}{x} enspace for enspace x < 0 end{Bmatrix}) = (begin{Bmatrix} 1 enspace for enspace x geq 0 -1 enspace for enspace x < 0 end{Bmatrix}) Here, (LHL = lim_{x to 0^-}(-1) = -1) (RHL = lim_{x to 0^+}(1) = 1) This shows LHL (neq) RHL. So, the limit of given function doesn't exists.

Prove that \(\lim_{x \to 0} \frac{|x|}{x}\) the does not exist.

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Prove that \(\lim_{x \to 0} \frac{|x|}{x}\) the does not exist.

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