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

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Solution:

Prove that lim x → 0 | x | x the does not exist. | CSIT Guide

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.

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