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I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ...
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:
In the above example, Kantorovitz shows that##\phi_0 (h) = - \frac{ \| h \|^2 }{( 1 + \sqrt{ 1 + \| h \|^2 )}^2 }##Kantorovitz then declares that ## \frac{ \phi_0 (h) }{ \| h \| } \rightarrow 0## as ##h \rightarrow 0## ... ...Can someone please show me how to demonstrate rigorously that this is true ... that is that
## \frac{ \phi_0 (h) }{ \| h \| } \rightarrow 0## as ##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:
Hope that helps readers understand the context and notation of the above post ,,, ,,,
Peter
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:
In the above example, Kantorovitz shows that##\phi_0 (h) = - \frac{ \| h \|^2 }{( 1 + \sqrt{ 1 + \| h \|^2 )}^2 }##Kantorovitz then declares that ## \frac{ \phi_0 (h) }{ \| h \| } \rightarrow 0## as ##h \rightarrow 0## ... ...Can someone please show me how to demonstrate rigorously that this is true ... that is that
## \frac{ \phi_0 (h) }{ \| h \| } \rightarrow 0## as ##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:
Peter
Attachments
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Kantorovitz - Example 4 ... Page 66 ... ... .png18.8 KB · Views: 869
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Kantorovitz - 1 - Sectiion on the DIfferential ... PART 1 ... .png27.4 KB · Views: 449
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Kantorovitz - 2 - Sectiion on the DIfferential ... PART 2 ... .png34.9 KB · Views: 436
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Kantorovitz - 3 - Sectiion on the DIfferential ... PART 3 ... .png13.2 KB · Views: 430
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