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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 $x\to a^-$. – Arturo Magidin Mar 26 '12 at 2:59
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? – ThisIsNotAnId Mar 26 '12 at 3:03
Above & below what? What is the space of values of $x$ here? – user2468 Mar 26 '12 at 3:09
@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 $$x\!\!\uparrow\!\!a = x\!\nearrow\!a = x\to a^-$$ – Arturo Magidin Mar 26 '12 at 3:13
Aha. Wikipedia is a badass. Planetmath mentions it as well. – user2468 Mar 26 '12 at 4:23
up vote 4 down vote accepted

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 $x→a^−$.

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