Browse Questions

# Evaluate $\lim\limits_{x \to 0} f(x)$ where $f(x)=\left\{\begin{array}{1 1}\large\frac{x}{|x|},&x\neq 0\\0,&x=0\end{array}\right.$

The given function is
$f(x)=\left\{\begin{array}{1 1}\large\frac{x}{|x|},&x\neq 0\\0,&x=0\end{array}\right.$
$= \lim\limits_{x \to 0^-}f(x) = \lim\limits_{x \to 0^-} \bigg[ \large\frac{x}{|x|} \bigg]$
$= \lim\limits_{ x \to 0} = \bigg[ \large\frac{x}{-x} \bigg]$$\quad \quad [ where\: x < 0, |X|= -x] = \lim\limits_{x \to 0} (-1) = -1 = \lim\limits_{x \to 0^+}f(x) = \lim\limits_{x \to 0^+} \bigg[ \large\frac{x}{|x|} \bigg] = \lim\limits_{ x \to 0} = \bigg[ \large\frac{x}{x} \bigg]$$ \quad \quad [ where\: x > 0, |X|= x]$
$= \lim\limits_{x \to 0} (1)$
$= 1$
It is observed thet $\lim\limits_{x \to 0^-} f(x) \neq \lim\limits_{x \to 0^+} f(x)$
Hence, $\lim\limits_{x \to 0} f(x)$ does not exist.