So today read something about two and three dimensional co-ordinate system and also about infinity and something came to me that I have been since pondering on.
So my question is, what is the area of an infinite line? Is it $\infty$ or is it$0$??.
Well if we consider the fact that it is one-dimentional, then of course area is $0$. But since we are mentioning it to be an infinite line, then area should be $\infty$, as it can be making a circle of infinite radius. Can anyone give me a technical explanation for this hypothetical question. Thanks, cheers!!
Best Answer
It really depends on the limiting process. Say you have a rectangle with length $x$ and width $1/x.$ You take $x\to \infty$ and this starts to look more and more like an infinite line since the length gets infinitely long and the width goes to zero. However, at all times the area is $1,$ which is neither zero, nor infinity.
On the other hand, you could take length $x^2$ and width $1/x$ and then the area would go to to infinity. Or you could take length $x$ and width $1/x^2$ and the area would go to zero. You could also take length $ax$ and width $1/x$ and make the area limit to $a$ for any $a>0.$