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I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard.

I am currently focused on Section 1.7: Derivatives in Several Variables as Linear Transformations ...

I need some help in order to understand some remarks by Hubbard and Hubbard on page 124 under the heading "The derivative in several variables ... ...

The relevant text reads as follows:

View attachment 8720

Referring to equation 1.7.10 H&H say the following:

" ... ... But this wouldn't work even in dimension \(\displaystyle 1\), because the limit changes sign as \(\displaystyle h\) approaches \(\displaystyle 0\) from the left and from the right. ... ... "Could someone please explain exactly how/why the limit changes sign as \(\displaystyle h\) approaches \(\displaystyle 0\) from the left and from the right. ... ... ?

I am puzzled because \(\displaystyle \mid h \mid\) doesn't change sign and \(\displaystyle f( a + h ) - f ( a)\) doesn't necessarily change sign as \(\displaystyle h\) approaches \(\displaystyle 0\) from the left and from the right. ... ...

Hope someone can help ...

Peter

I am currently focused on Section 1.7: Derivatives in Several Variables as Linear Transformations ...

I need some help in order to understand some remarks by Hubbard and Hubbard on page 124 under the heading "The derivative in several variables ... ...

The relevant text reads as follows:

View attachment 8720

Referring to equation 1.7.10 H&H say the following:

" ... ... But this wouldn't work even in dimension \(\displaystyle 1\), because the limit changes sign as \(\displaystyle h\) approaches \(\displaystyle 0\) from the left and from the right. ... ... "Could someone please explain exactly how/why the limit changes sign as \(\displaystyle h\) approaches \(\displaystyle 0\) from the left and from the right. ... ... ?

I am puzzled because \(\displaystyle \mid h \mid\) doesn't change sign and \(\displaystyle f( a + h ) - f ( a)\) doesn't necessarily change sign as \(\displaystyle h\) approaches \(\displaystyle 0\) from the left and from the right. ... ...

Hope someone can help ...

Peter