Abstract:
By comparing two methods of extension of real numbers, that is, the affine extension, and the projective extension, together with studying and summarizing the advantages and drawbacks when introducing, in real analysis, especially in the limit theory of real-value functions, the signed infinities
- \infty , + \infty and the unsigned infinity
\infty. It is obtained that, without introduction of
\mathop \lim \limits_x \to x_\rm0 f(x) = \infty , there are, in real analysis, no confusions of notation for limit expression any longer, and the limit theory of real-valued functions is more precise, by just using only
+\infty and
-\infty, and hence, the depiction of local properties of functions near the limit point is easier and more precise. Moreover, without unsigned infinity, the whole limit theory of real-valued functions is more clear and unified, which make it possible to generalize some important theorems in real analysis, with neat proofs and multitude of applications.