MHB What is the connection between sin and cos in Example 6.3.4?

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I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...

I am focused on Chapter 6: Differentiation ...

I need help in fully understanding the an aspect of Example 6.3.4 ...Example 6.3.4 ... reads as follows:View attachment 7305The above example implies that:

$$\frac{ \text{ sin } x}{ \sqrt{x} } = \frac{ \text{ cos } x}{ 1/ 2 \sqrt{x} } $$
Can someone please explain how/why this is true ...Peter
 
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Peter said:
The above example implies that:

$$\frac{ \text{ sin } x}{ \sqrt{x} } = \frac{ \text{ cos } x}{ 1/ (2 \sqrt{x}) } $$
Can someone please explain how/why this is true ...
No, the example does not imply that those two functions are the same. It just says that (by applying l'Hospital's rule) they have the same limit as $x\to0+$.
 
Opalg said:
No, the example does not imply that those two functions are the same. It just says that (by applying l'Hospital's rule) they have the same limit as $x\to0+$.
Oh! Of course ... how silly of me ...

Thanks 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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