What do the $+,-$ mean in limit notation, like$\lim\limits_{t \to 0^+}$ and $\lim\limits_{t \to 0^-}$?

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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.

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4 Answers

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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

$$\lim_{x \to 0^+}H(x)=1$$ $$\lim_{x \to 0^-}H(x)=0$$

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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.

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And a good, visual example could be something like

                            

$$\begin{align} \lim_{x\to x_0^-}f&=y_1\\ \lim_{x\to x_0^+}f&=y_2 \end{align}$$

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$$\lim_{t \to 0^{+}}$$ 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|) $

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