What is the limit value of $\frac{sin(x)}{|x|}$ as $x\rightarrow 0?$
I tried to solve this question by dividing it into two cases $\mathrm{(i)} \ x>0,$ and $\mathrm{(ii)}\ x<0.$ When $x>0,$ the limit value goes to one, as we know.
But, when $x<0$ the absolute of x becomes $-x.$ So, the limit is -1.
So i conclude it that it doesn't have limit value because when $x>0$ it has 1 and when $x<0$ it has -1.
Am i correct, or there is a mistake?
From wikipedia:
"Alternatively x may approach p from above (right) or below (left), in which case the limits may be written as
$$ \lim _{x\to p^{+}}f(x)=L $$
or
$$\lim _{x\to p^{-}}f(x)=L$$
respectively. If these limits exist at p and are equal there, then this can be referred to as the limit of f(x) at p. If the one-sided limits exist at p, but are unequal, there is no limit at p (the limit at p does not exist). If either one-sided limit does not exist at p, the limit at p does not exist."
Therefore you are correct that the limit of $\sin(x) / \text{abs}(x)$ at $x=0$ does not exist.