*Originally posted by Fwack*

What is the limit of x approaching 0 of 1/x?

Correct answer; it has no limit. Though often one says that the limit is infinity. If you calculate 1/x for small x, you'll never get to infinity.

Although this is true, strictly speaking, you can give the answer as infinity. It's not technically true, but because 1/x can be made arbitrarily large with very small values of x, we pretend that it's infinity.

If the right handed limit is not the same as the left handed limit in 1/x when x approaches zero is not the same, there is no limit, even if the two values belongs to R.

In the case of lim 1/abs(x) the two limits are the same but nevertheless not within R, there isn't any limit (within R).

One should never consider the infinity within R, because it isn't. If we define R as one, R*, where there is an infinit value, oo, we get very mysterious results, like oo+1=oo, oo*2=oo, oo^2=oo, even oo^oo=oo, very confusing. It seems that R* demands arithmetic properties you can't use in ordinary R.

Every time anyone says that 0/0 has a value, that you acutally perform a division with zero in sin(x)/x = 1, than you show that you have not any deeper insights in mathematics.

You are absolutely right, Fwack, I just expand the issue a little.