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I am currently focused on Chapter 2: Derivation ... ...

I need help with an aspect of Kantorovitz's Example 4 on page 66 ...

Kantorovitz's Example 4 on page 66 reads as follows:View attachment 7817In the above example, Kantorovitz show that\(\displaystyle \phi_0 (h) = - \frac{ \| h \|^2 }{( 1 + \sqrt{ 1 + \| h \|^2 )}^2 }\)Kantorovitz then declares that \(\displaystyle \frac{ \phi_0 (h) }{ \| h \| } \rightarrow 0\) as \(\displaystyle h \rightarrow 0\) ... ...Can someone please show me how to demonstrate rigorously that this limit is as stated i.e that is that \(\displaystyle \frac{ \phi_0 (h) }{ \| h \| } \rightarrow 0\) as \(\displaystyle h \rightarrow 0\) ... ...

... ... Help will be much appreciated ...

Peter============================================================================================

***NOTE***

Readers of the above post may be helped by having access to Kantorovitz' Section on "The Differential" ... so I am providing the same ... as follows:View attachment 7818

View attachment 7819

https://www.physicsforums.com/attachments/7820