What do the $+,-$ mean in limit notation, like$\lim\limits_{t \to 0^+}$ and $\lim\limits_{t \to 0^-}$? I'm working on Laplace Transforms and have got to a section where they are talking about zero to the power plus or minus and that they are different. I can't remember what this means though.
It's generally used in limits.
$\lim\limits_{t\to 0^-}$ or $\lim\limits_{t\to 0^+}$
Any help would be much appreciated.
 A: And a good, visual example could be something like
                            
$$\begin{align}
\lim_{x\to x_0^-}f&=y_1\\
\lim_{x\to x_0^+}f&=y_2
\end{align}$$
A: 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
$$\lim_{x \to 0^+}H(x)=1$$
$$\lim_{x \to 0^-}H(x)=0$$
A: In this case, the plus and minus refer to the direction from which you approach zero.  So, 
$\lim \limits_{t \to 0^{-}}$
means the limit as $t$ approaches $0$ from the negative side, or from below, while
$\lim \limits_{t \to 0^{+}}$
means the limit as $t$ approaches $0$ from the possitive side, or from above.
So, it is just specifying which direction you are moving along the number line.
A: $$\lim_{t \to 0^{+}}$$ indicates that the limit is meant to be taken only from the positive direction; it's a one-sided limit
Right hand Limit: $\displaystyle\lim_{t \to 0+} f(x) = \displaystyle\lim_{t \to 0} f(x+|t|) $.
Left Hand Limit: $\displaystyle\lim_{t \to 0-} f(x)= \displaystyle\lim_{t \to 0} f(x-|t|)  $
