Proving that a limit does not exist Given the function 
$$f(x)= \left\{\begin{matrix}
  1 &amp; x \gt 0 \\
  0 &amp; x =0 \\
  -1 &amp; x \lt 0
 \end{matrix}\right\}$$

What is $\lim_{a}f$ for all $a \in \mathbb{R},a \gt 0$?

It seems easy enough to guess that the limit is $1$, but how do I take into account the fact that $f(x)=-1$ when $x \lt 0$?
Thanks
For $a&gt;0$, let $\delta=a$. Then for any $\epsilon&gt;0$, we have that 
$$|x-a|&lt\delta\implies x&gt;0\implies f(x)=1\implies |f(x)-1|&lt\epsilon$$
hence $\lim\limits_{x\to a}f(x)=1$, as we have satisfied the definition of limit.
Here $\lim_{x\to 0^+}f(x)=1$ &amp; $\lim_{x\to 0^-}f(x)=-1$ both limit are not equal.

therefore limit is not exists