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

I need help with an aspect of Kantorovitz's Example 3 on pages 65-66 ...

Kantorovitz's Example 3 on pages 65-66 reads as follows:

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In the above example, we read the following:"... ... \(\displaystyle \frac{ \mid \phi_0 (h) \mid }{ \| h \| } = \frac{ \mid h_1 \text{ sin } (h_2 h_3) \mid }{ \| h \|^{ a + 1 } }\) ... ... ... "

My question is as follows:In the Section on The Differential (see scanned text below) ...

Kantorovitz defines \(\displaystyle \phi_x(h)\) as follows:

\(\displaystyle \phi_x(h) := f(x +h) - f(x) - Lh\)

so that

\(\displaystyle \phi_0(h) := f(0 +h) - f(0 ) - Lh = f(h) - f(0)\) ...... BUT in the Example ... as I understand it ... \(\displaystyle f(0)\) does not exist ...? ...

... BUT ... Kantorovitz effectively gives \(\displaystyle \mid \phi_0 (h) \mid = \frac{ \mid h_1 \text{ sin } (h_2 h_3) \mid }{ \| h \| } \)

Can someone please explain how Kantorovitz gets this value when \(\displaystyle f(0)\) does not exist?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:

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