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$\begingroup$

I'm wondering what is the difference in the use of

$$\lim\limits_{x \downarrow a}$$

$$\lim\limits_{x \searrow a}$$ $$\lim\limits_{x \nearrow a}$$

$$\lim\limits_{x \uparrow a}$$

I see them around and I don't know what they really mean. Do the arrows characterize how $x$ tends to $a$?

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    $x\downarrow a$ means that $x$ is approaching $a$ "from above", in a decreasing manner; it's much like $x\to a^+$, "approaching from the right"; same for $x\searrow a$. $x\uparrow a$ means $x$ approaches $a$ from below, in an increasing manner, much like $x\to a^-$.2012-03-26
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    I would suppose. Since the usual $\rightarrow$ implies the variable tends to the (finite) constant from both ends. In any case, I think taking a guess works out well usually. What does the context say?2012-03-26
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    Above & below what? What is the space of values of $x$ here?2012-03-26
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    @J.D. Presumably, the real numbers, with "above" meaning "from values greater", and "below" meaning "from values smaller"; ie.., $$x\!\!\downarrow\!\!a = x\!\searrow\!a = x\to a^+$$and $$x\!\!\uparrow\!\!a = x\!\nearrow\!a = x\to a^-$$2012-03-26
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    Aha. [Wikipedia is a badass](http://en.wikipedia.org/wiki/One-sided_limit). Planetmath [mentions it](http://planetmath.org/jsmath/jsMath-global.html?http%3A//planetmath.org/encyclopedia/OneSidedLimit.html) as well.2012-03-26
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    I am still not a hundred percent sure whether I understand the difference. Could you give an example of a sequence $(x_n)_n$ for which $\lim_{n \to \infty} x_n \neq \lim_{n \uparrow \infty} x_n$?2017-04-13

1 Answers 1

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Arturo Magidin:

$x↓a$ means that $x$ is approaching a "from above", in a decreasing manner; it's much like $x→a^+$, "approaching from the right"; same for $x↘a$. $x↑a$ means $x$ approaches a from below, in an increasing manner, much like $x→a^−$.