The Riemann Sphere is another imagination of the complex plane. In that imagination infinity is represented by the top pole of the sphere. Therefore it is imagined as only one single point.
But when we (in our lessons) deal with real numbers only, we image there is a +∞ and a -∞ (for example when talking about limits). Therefore my two questions: 1. Why is there no Riemann Circle for real numbers only? 2. Why is infinity in the real numbers not imagined as one point?
If you allow the Riemann sphere then in it you could say you have the real line with one point at infinity.
However in Calculus $\infty$ and $-\infty$ are not really numbers or places. It is a convenient notation because some of what it makes you think you can do is correct, but you have to check.
The way to read $\lim_{x \rightarrow 1}f(x)=\infty$ is NOT "$f(x)$ converges to $\infty$." we say "$f(x)$ grows without bound (as $x$ goes to $1$)" or "$f(x)$ diverges to $\infty$."
However $\lim_{x \rightarrow 1}f(x)=\infty$ does suggest that $\lim_{x \rightarrow 1}\frac1{f(x)}=0.$ It is then easy to prove that is correct.
Also, we want to say $\lim_{x \rightarrow \infty} \arctan(x)=\frac{\pi}2$ while $\lim_{x \rightarrow -\infty} \arctan(x)=-\frac{\pi}2.$
