# How should I interpret a plus superscript in limit notation?

I am doing some calc refresher problems and I found this notation...

How should I interpret that? As x approaches positive two? What does the + mean?

This is called the right hand limit, and $x$ is said to approach $2$ from the right. The definition of the limit is modified accordingly:

$$\left ( \forall \varepsilon >0 \right ) \left (\exists \delta >0 \right ) \left ( 0<x-2<\delta \implies |f(x)-f(2)|<\varepsilon \right )$$

• Thanks @Reveillark. I know the ∀ and ∃ notation but what does the ⟹ mean? Can you rephrase that in English? part of what I want to learn is how to read different calc notations. Commented Apr 28, 2015 at 1:40
• It means 'implies'. In English it would be: $0<x-2<\delta$ implies that $|f(x)-f(2)|<\varepsilon$. Alternatively: If $0<x-2<\delta$ then $|f(x)-f(2)|<\varepsilon$. Commented Apr 28, 2015 at 1:47

That means $x$ aproaching 2 from the right. That is $x$ aproaching 2 but greater than 2. It does not make sense to say x aproaches positive 2 at least for me.

Yes, as x approaches positive 2, or just 2, from the right side.

$\lim\limits_{x\to c^+} f(x)$ is interpreted to mean "the limit of $f(x)$ as $x$ approaches $c$ from the right side".

In particular, you should be aware that $\lim\limits_{x\to c}f(x)$ exists if and only if both $\lim\limits_{x\to c^+}f(x)$ and $\lim\limits_{x\to c^-}f(x)$ both exist and are equal. (where $\lim\limits_{x\to c^-}f(x)$ is the limit as $x$ approaches $c$ on the left side).

Example: $\lim\limits_{x\to 1^+} \lfloor x\rfloor = 1$ while $\lim\limits_{x\to 1^-} \lfloor x\rfloor = 0$, so the limit $\lim\limits_{x\to 1} \lfloor x\rfloor$ does not exist.

(where $\lfloor x\rfloor$ denotes the floor function, i.e. the greatest integer less than or equal to $x$)