You are overloading the word “real”. In terms of the mathematics of the reals, there is no difference. In terms of real-life pragmatics, 0 and -0 could be used to differentiate between different outcomes in a way that is sometimes useful.

Infinity and -0 are not in the real numbers, so the expression there makes no sense if you are thinking strictly in the real numbers. If you assign real number bounds to what the floating point numbers mean, the expressions make sense.

In floating point terms infinity tends to indicate overflow (any number that is too big to represent) and the 0's indicate underflows. So in more verbose terms,

1/(positive overflow) = (positive underflow)

While

-1/(positive overflow) = (negative underflow)

In this case, since the positive overflow isn't really infinity and the underflows aren't really 0, they are not equal. In practice, -0 and the 0 can both also arise from situations that produce 0 exactly, too, but this is not that case.

You may be thinking about how lim{x->inf}(1/x) = 0 = lim{x->inf}(-1/x), which is true. Infinity in floating point does not necessarily represent that limit, though, just any number that is too big.

You may also notice that the limit is not in the range of the functions inside the limit. For all real x, 1/x != 0

We are saying the same thing.