Question

lim ((x/|x|) +1)
x➡0​

Answer 1

First let us find left hand limit.

For $\displaystyle\sf {x\leq 0,}$

$\displaystyle\sf{\longrightarrow |x|=-x}$

$\displaystyle\sf{\longrightarrow \dfrac {x}{|x|}=-1}$

$\displaystyle\sf{\longrightarrow \dfrac {x}{|x|}+1=0}$

Therefore,

$\displaystyle\sf{\longrightarrow\lim_{x\to 0^-}\left (\dfrac {x}{|x|}+1\right)=0\quad\quad\dots(1)}$

Now let us find right hand limit.

For $\displaystyle\sf {x\geq 0,}$

$\displaystyle\sf{\longrightarrow |x|=x}$

$\displaystyle\sf{\longrightarrow \dfrac {x}{|x|}=1}$

$\displaystyle\sf{\longrightarrow \dfrac {x}{|x|}+1=2}$

Therefore,

$\displaystyle\sf{\longrightarrow\lim_{x\to 0^+}\left (\dfrac {x}{|x|}+1\right)=2\quad\quad\dots(2)}$

Comparing (1) and (2),

$\displaystyle\sf{\longrightarrow \lim_{x\to 0^-}\left (\dfrac {x}{|x|}+1\right)\neq\lim_{x\to 0^+}\left (\dfrac {x}{|x|}+1\right)}$

Hence the limit does not exist.