Fraction Limit

Relevant wiki: L'Hopital's Rule - Basic

$\begin{aligned} L_+ & = \lim_{x \to 0^+} \frac {|x|}x \\ & = \lim_{x \to 0^+} \frac xx & \small \color{#3D99F6} \text{A 0/0 case, L'Hôpital's rule applies.} \\ & = \lim_{x \to 0^+} \frac 11 & \small \color{#3D99F6} \text{Differentiate up and down w.r.t. }x \\ & = 1 \end{aligned}$

$\begin{aligned} L_- & = \lim_{x \to 0^-} \frac {|x|}x \\ & = \lim_{x \to 0^-} \frac {-x}x & \small \color{#3D99F6} \text{A 0/0 case, L'Hôpital's rule applies.} \\ & = \lim_{x \to 0^-} \frac {-1}1 & \small \color{#3D99F6} \text{Differentiate up and down w.r.t. }x \\ & = -1 \end{aligned}$

Since $L_+ \ne L_-$ , the limit does not exist .