We have f (x) = $\begin{cases} |x| + 1, & x < 0 \\ 0, & x = 0 \\ |x| - 1, & x > 0 \end{cases}$ Case (i) : When a < 0, we have $\lim\limits_{x \to a}$ f (x) = $\lim\limits_{x \to a^{-}}$ |x| + 1 = $\lim\limits_{x \to a^{-}}$ (- x + 1) = - a + 1 Case (ii) : When a > 0, we have $\lim\limits_{x \to a}$ f (x) = $\lim\limits_{x \to a^{+}}$ (x - 1) = a - 1 Case (iii) : When a = 0, we have = $\lim\limits_{x \to a^{+}}$ f (x) = $\lim\limits_{x \to 0^{+}}$ f (x) = $\lim\limits_{x \to 0^{+}}$ (|x| - 1) = 0 - 1 = - 1 $\lim\limits_{x \to a^{-}}$ = $\lim\limits_{x \to 0^{-}}$ f (x) = $\lim\limits_{x \to 0^{-}}$ (|x| + 1) = - 0 + 1 = 1 ∴ $\lim\limits_{x \to a^{+}}$ f (x) ≠ $\lim\limits_{x \to a^{-}}$ f (0) [Since f (0) = 0] $\lim\limits_{x \to a}$ f (x) does not exist when a = 0. ∴ $\lim\limits_{x \to a}$ f (x) exists for all a ≠ 0.