Show that a circle on the sphere that does not pass through the north pole corresponds to a circle in the complex plane.

Suppose that \(J: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}\) is defined by \(J(z)=1 / z, z \in \mathbb{C}_{\infty} .\) Do our conventions imply \(J(0)=\infty\) and \(J(\infty)=\infty\) ? Does $$ \chi(J(z), J(w))=\chi(z, w) $$ hold in \(\mathbb{C}_{\infty} ?\)

Which of the following sequences are convergent? (a) \(\left\\{i^{n}\right\\}\) (b) \(\left\\{z_{0}^{n}\right\\}\), where \(\left|z_{0}\right|<1\) (c) \(\left\\{\frac{\cos n+i \sin n}{n}\right\\}\) (d) \(\left\\{\frac{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}}{n}\right\\}\) (f) $\left\\{e^{n \pi i / 3}+\left(-\frac{1}{2}-i \frac{\sqrt{3}}{2}\right)^{n}\right\\}$ (g) $\left\\{e^{n \pi i / 6}+\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{n}\right\\}$ (f) \(\left\\{\frac{n \cos (n \pi)}{2 n+1}\right\\}\) (i) \(\left\\{\sin \left(\frac{n \pi}{8}\right)\right\\}\).

If \(\lim _{z \rightarrow \infty} f(z)=a\), and \(f(z)\) is defined for every positive integer \(n\), prove that \(\lim _{n \rightarrow \infty} f(n)=a\). Give an example to show that the converse is false.

Give an example of a sequence that (a) does not converge, but has exactly one limit point; (b) has \(n\) limit points, for any given integer \(n\); (c) has infinitely many limit points.

Find the following limits: (a) \(\lim _{z \rightarrow 0} f(z)\), where $f(z)=\frac{x y}{x^{2}+y^{2}}+2 x i$, (b) \(\lim _{z \rightarrow 0} f(z)\), where $f(z)=\frac{x y}{x^{2}+y}+2 \frac{x}{y} i$, (c) \(\lim _{z \rightarrow 0} f(z)\), where $f(z)=\frac{x y^{3}}{x^{3}+y^{3}}+\frac{x^{8}}{y^{2}+1} i$.