MHB Limits of Complex Functions .... Final Remark from Palka in Section 2.2 ....

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I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ...

I am focused on Chapter 2: The Rudiments of Plane Topology ...

I need help with some aspects regarding an example in Palka's final remarks in Section 2.2 Limits of Functions ...

Palka's final remarks in Section 2.2 which include the example read as follows:View attachment 7372I have two questions regarding the above text ...Question 1

In the above text from Palka Section 2.2 we read the following:" ... ... Consider for a moment, the function $$f(x) = e^{-1/x^2}$$. We are aware from calculus that $$\text{lim}_{x \rightarrow 0} \ f(x) = \text{lim}_{x \rightarrow 0} \ e^{-1/x^2} = 0$$. ... ..."Can someone please demonstrate exactly how/why $$\text{lim}_{x \rightarrow 0} \ f(x) = \text{lim}_{x \rightarrow 0} \ e^{-1/x^2} = 0$$ ... ... ?Question 2

In the above text from Palka Section 2.2 we read the following:

" ... ... However, if we set $$z_n = (2n \pi)^{-1/2} e^{ \pi i / 4 }$$ for $$n = 1,2, \ ... $$ , we observe that $$z_n \rightarrow 0$$, whereas $$f(z_n) = e^{ 2n \pi i } = 1$$ for every $$n$$ so $$f( z_n ) \rightarrow 1$$. ... ... "I am unable to show that $$f(z_n) = e^{ 2n \pi i }$$ ... can someone please help ...Help will be much appreciated ...

Peter
 
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Peter said:
Question 1

Can someone please demonstrate exactly how/why $$\text{lim}_{x \rightarrow 0} \ f(x) = \text{lim}_{x \rightarrow 0} \ e^{-1/x^2} = 0$$ ... ... ?
Starting from the fact that $e^t \to \infty$ as $t\to\infty$, it follows that $\lim_{t\to\infty}e^{-t} = \lim_{t\to\infty}\dfrac1{e^t} = \dfrac1{\lim_{t\to\infty}e^{t}} = 0.$ If you then put $t = 1/x^2$, noticing that $t\to\infty$ as $x\to0$, it follows that $\lim_{x\to0}e^{-1/x^2} = 0.$

Peter said:
Question 2

In the above text from Palka Section 2.2 we read the following:

" ... ... However, if we set $$z_n = (2n \pi)^{-1/2} e^{ \pi i / 4 }$$ for $$n = 1,2, \ ... $$ , we observe that $$z_n \rightarrow 0$$, whereas $$f(z_n) = e^{ 2n \pi i } = 1$$ for every $$n$$ so $$f( z_n ) \rightarrow 1$$. ... ... "I am unable to show that $$f(z_n) = e^{ 2n \pi i }$$ ... can someone please help ...
If $$z_n = (2n \pi)^{-1/2} e^{ \pi i / 4 }$$ then $$z_n^2 = (2n \pi)^{-1} e^{ \pi i / 2 }$$. But $e^{ \pi i / 2 } = i$. So $z_n^2 = \dfrac i{2n\pi}$, and $\dfrac1{z_n^2} = \dfrac{2n\pi}i = -2n\pi i.$ Finally, $$f(z_n) = e^{-1/z_n^2} = e^{ 2n \pi i } = 1.$$
 
Opalg said:
Starting from the fact that $e^t \to \infty$ as $t\to\infty$, it follows that $\lim_{t\to\infty}e^{-t} = \lim_{t\to\infty}\dfrac1{e^t} = \dfrac1{\lim_{t\to\infty}e^{t}} = 0.$ If you then put $t = 1/x^2$, noticing that $t\to\infty$ as $x\to0$, it follows that $\lim_{x\to0}e^{-1/x^2} = 0.$If $$z_n = (2n \pi)^{-1/2} e^{ \pi i / 4 }$$ then $$z_n^2 = (2n \pi)^{-1} e^{ \pi i / 2 }$$. But $e^{ \pi i / 2 } = i$. So $z_n^2 = \dfrac i{2n\pi}$, and $\dfrac1{z_n^2} = \dfrac{2n\pi}i = -2n\pi i.$ Finally, $$f(z_n) = e^{-1/z_n^2} = e^{ 2n \pi i } = 1.$$
Thanks for the help, Opalg ...

Peter
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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