Say we let
$H(x)=\begin{cases} 0, & x < 0, \\ 1, & x > 0, \end{cases}$
and let $H(0)$ be not defined.
Say I would like to approach $0$ on this function. However, a problem arises! Looking at the plot of the function, it is clear that if one were to approach from the right hand side, the limit is $1$, whilst if one approaches from the left, the limit is $0$ and thus the two-sided limit does not exist (both sides should be approaching the same number for this limit to exist)! This can also be easily seen by plugging in numbers:
$H(1)=1$ $H(.1)=1$ $H(.000000000001)=1$ etc. But, doing the same thing from the left hand side, we find
$H(-1)=0$ $H(-.1)=0$ $H-(.000000000001)=0$

Thus we need to define a different type of limit for functions with similar discontinuities so we may approach from either side. This limit is the "one-sided limit" and is used generally when a two-sided limit does not exist, like in the above case. $\lim_{x \to x_0^+}f(x)$ represents the right handed limit of $f(x)$ to $x_0$ whilst $\lim_{x \to x_0^-}f(x)$ represents the left hand limit. So we see that $\lim_{x \to 0} H(x)$ does not exist, but
$\lim_{x \to 0^+}H(x)=1$ $\lim_{x \to 0^-}H(x)=0$