Draw the graph of function $f (x) = | x | / x .$ Is f(x) defined at $x = 0 ?$ Does the limit of $f (x)$ exist when $x \to 0$ ?

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Solution

$f (x) = \frac{| x |}{x}$

$f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{x}{x} = 1 x > 0 \frac{0}{x} = 0 x = 0 \frac{- x}{x} = - 1 x < 0 \end{matrix}$

Its clear from the graph and above definition, f(x) is defined at $x = 0$
Also, for all positive real values of $x$ , f(x) gives value 1 and for all negative real values of $x$ , $f (x)$ gives value -1.

So, ${lim}_{x \to 0} f (x)$ does not exists.

We can also show it mathematically,
$L H L = {lim}_{x \to 0^{-}} f (x)$
$\Rightarrow L H L = - 1$

$R H L = {lim}_{x \to 0^{+}} f (x)$
$\Rightarrow R H L = 1$

Since, $R H L \neq L H L$ .
Hence, limit of f(x) does not exist at $x = 0$

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