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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in fully understanding some remarks by Browder made after Definition 8.9 ...

Definition 8.9 and the following remark read as follows:

View attachment 9408

In the above Remark by Browder we read the following:

"for any fixed \(\displaystyle k \neq 0\) and \(\displaystyle t \gt 0\), we have \(\displaystyle \frac{1}{ |tk| }( L(tk) - M(tk) ) = \frac{1}{|k|},(Lk - Mk )\) ... ... ... "

My questions are as follows:

Browder puts \(\displaystyle h = tk\) and then let's \(\displaystyle t \to 0\) ... why is Browder doing this ... what is the logic behind this ... what do we gain by putting \(\displaystyle h = tk\) ... both \(\displaystyle h\) and \(\displaystyle k \in \mathbb{R}^n \) and also isn't \(\displaystyle h\) just as arbitrary as \(\displaystyle k\) ... ?

How exactly (and in detail) does letting \(\displaystyle t \to 0\) allow us to conclude that \(\displaystyle Lk = Mk\) ...

Help will be much appreciated ...

Peter

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in fully understanding some remarks by Browder made after Definition 8.9 ...

Definition 8.9 and the following remark read as follows:

View attachment 9408

In the above Remark by Browder we read the following:

"for any fixed \(\displaystyle k \neq 0\) and \(\displaystyle t \gt 0\), we have \(\displaystyle \frac{1}{ |tk| }( L(tk) - M(tk) ) = \frac{1}{|k|},(Lk - Mk )\) ... ... ... "

My questions are as follows:

**Question 1**Browder puts \(\displaystyle h = tk\) and then let's \(\displaystyle t \to 0\) ... why is Browder doing this ... what is the logic behind this ... what do we gain by putting \(\displaystyle h = tk\) ... both \(\displaystyle h\) and \(\displaystyle k \in \mathbb{R}^n \) and also isn't \(\displaystyle h\) just as arbitrary as \(\displaystyle k\) ... ?

**Question 2**How exactly (and in detail) does letting \(\displaystyle t \to 0\) allow us to conclude that \(\displaystyle Lk = Mk\) ...

Help will be much appreciated ...

Peter