We have
lim x → 0 + x | tan x | = lim x → 0 + x tan x = 1 , {\displaystyle \lim _{x\rightarrow 0^{+}}{\frac {x}{|\tan x|}}=\lim _{x\rightarrow 0^{+}}{\frac {x}{\tan x}}=1,} but
lim x → 0 − x | tan x | = lim x → 0 − x − tan x = − 1 , {\displaystyle \lim _{x\rightarrow 0^{-}}{\frac {x}{|\tan x|}}=\lim _{x\rightarrow 0^{-}}{\frac {x}{-\tan x}}=-1,} so the limit does not exist {\displaystyle \color {blue}{\text{the limit does not exist}}} .