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I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Section 3.1 Partial Derivatives and Directional Derivatives ...

I need some help with Example 3 in Chapter 3, Section 1 ...

Example 3 in Chapter 3, Section 1 reads as follows:View attachment 8587In the above text we read the following:" ... ... \(\displaystyle \lim_{ t \to 0 } \frac{ \| a + t \frac{a}{ \| a \| } \| - \| a \| }{ t } = \lim_{ t \to 0 } \frac{ ( \| a \| - t ) - \| a \| }{ t } \) ... ... "

Can someone please explain exactly how \(\displaystyle \lim_{ t \to 0 } \frac{ \| a + t \frac{a}{ \| a \| } \| - \| a \| }{ t } = \lim_{ t \to 0 } \frac{ ( \| a \| - t ) - \| a \| }{ t } \) ... ... ... indeed, specifically how \(\displaystyle \| a + t \frac{a}{ \| a \| } \| = ( \| a \| - t )\) ... ...

Help will be appreciated ...

Peter

I need some help with Example 3 in Chapter 3, Section 1 ...

Example 3 in Chapter 3, Section 1 reads as follows:View attachment 8587In the above text we read the following:" ... ... \(\displaystyle \lim_{ t \to 0 } \frac{ \| a + t \frac{a}{ \| a \| } \| - \| a \| }{ t } = \lim_{ t \to 0 } \frac{ ( \| a \| - t ) - \| a \| }{ t } \) ... ... "

Can someone please explain exactly how \(\displaystyle \lim_{ t \to 0 } \frac{ \| a + t \frac{a}{ \| a \| } \| - \| a \| }{ t } = \lim_{ t \to 0 } \frac{ ( \| a \| - t ) - \| a \| }{ t } \) ... ... ... indeed, specifically how \(\displaystyle \| a + t \frac{a}{ \| a \| } \| = ( \| a \| - t )\) ... ...

Help will be appreciated ...

Peter

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