@rjclayton:

So Meijing, this is nothing but poor notation?

I decided to elaborate, why x -> -0 is poor notation.

First let’s see what this notation is supposed to mean.

The limit expresses the behavior of a function as its parameter approaches a given value. The function defined by f(x) = (x*x)/x + 1 is not defined for x=0 but f(x) = x + 1 for any other value of x. If x is close to 0, f(x) = x + 1 is close to 1. This behavior is written as f(x) -> 1 for x -> 0 or lim(x -> 0)f(x) = 1.

As an other example we will pick the sign function, which is defined by

sign(x) = -1 if x is negative

sign(x) = +1 if x is positive

sign(0) = 0 (which is just mentioned for completeness, but isn’t relevant in this context)

In this case there is no definite behavior for values close to 0. For any x (not equal to 0), no matter how close it is to 0, sign(x) can be +1 or -1. sign(x) does not approach a definite value.

But if we only consider, such values close to 0, which are smaller than 0. There is a definite behavior. For these values sign(x) = -1. If we consider values close to 0, which are greater than 0, sign(x) = +1. There is some definite behavior of sign around 0, but only if we distinguish between approaching the number from the left and approaching it from the right.

x -> -0 is supposed to mean approach 0 from the left. But this is poor notation, as it not possible to use it for other numbers but 0. x -> -1 doesn’t mean approaching -1 (or 1) from the left, but approaching -1 from any side.

A better notation would be x -> 0-, as this notation can also be used for -1 and +1 (x -> -1- an x -> 1-). There are other notations which are even better but impossible to write using BBCode (I hope this link will continue to work).