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

[Math Amateur]

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

 

[Opalg]

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 $x\to0+$.

 

[Math Amateur]

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