\[\lim_{x \rightarrow a}f \left( x \right)=c \iff \lim_{x \rightarrow a ^{+}}f \left( x \right)=\lim_{x \rightarrow a ^{-}}f \left( x \right)=c\]
This says the limit exists if and only if the limit from the left and right are the same.
You can have different left and right limits, in which case the overall limit will not exist.
For example, let \[f \left( x \right)=\frac{ 1 }{ \left( x-3 \right) }\]
in order to determine \[\lim_{x \rightarrow 3}f \left( x \right)\] we need to look at what happens on the left and right as x approaches 3.
\[\lim_{x \rightarrow 3^{+}}\frac{ 1 }{ \left( x-3 \right) }=+\infty\]because \[\left( x-3 \right)>0, \forall x>3\].
Likewise, \[\lim_{x \rightarrow 3^{-}}\frac{ 1 }{ \left( x-3 \right) }=-\infty\]because \[\left( x-3 \right)<0, \forall x<3\].
The bottom is going to 0 as x gets closer and closer to 3. But coming from above 3 (like 3.1, 3.01, 3.001), the bottom is still positive as it goes to zero. yetcoming from below 3 (like 2.9, 2.99, 2.999) the bottom is still negative as it goes to zero.
I hope that helps cause it was alot to type.