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

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.

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