Notation or whatever you want to call it

[rocomath]

Example:

$$\lim_{x \rightarrow 0^+}|x|$$

$$x \geq 0$$

$$\lim_{x \rightarrow 0^-}|x|$$

$$x < 0$$

now for a similar limit, such as

$$\lim_{x \rightarrow -4^+}|x+4|$$

the Domain is from

$$x > -4$$

Why isn't it like |x|, whose Domain is greater than or equal to 0? Which in this case, greater than or equal to -4.

 

[Gib Z]

What you typed doesn't really make much sense :( Really you just wrote some lines of tex and expected us to follow >.< They mean nothing on their own.

 

[HallsofIvy]

Example:

$$\lim_{x \rightarrow 0^+}|x|$$

$$x \geq 0$$

For $x\ge 0$ |x| is just x

$$\lim_{x \rightarrow 0^-}|x|$$

$$x < 0$$

For $x< 0$ |x| is just -x

now for a similar limit, such as

$$\lim_{x \rightarrow -4^+}|x+4|$$

the Domain is from

$$x > -4$$

Why isn't it like |x|, whose Domain is greater than or equal to 0? Which in this case, greater than or equal to -4.

Not unless it is specified to be that. The domain for |x+4|, just like the domain for |x| is "all real numbers". Of course |y| is y if $y\ge 0$, -y if y< 0. Letting y= x+4, |x+4|= x+4 if $x+4\ge 0$ which is the same as $x\ge -4$, and |x+4|= -(x+4) if x+4< 0 which is the same as x< -4. That's why it is convenient sometimes to restrict |x| to $x\ge 4$ or |x+4| to $x\ge -4$.