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*m*[/FONT] to [FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Math]

*n*[/FONT] ... ...

I need help in order to fully understand exactly how/why equations (3.4) and (3.5) follow ... ...

The relevant portion of Tapp's text reads as follows:

My questions are as follows:

**Question 1**In the above text from Tapp we read the following:

" ... ... If the directional derivative \(\displaystyle df_p(v) = \displaystyle \lim_{ t \to 0 } \frac{ f(p + tv) - f(p) }{t}\) exists, then the limit can be reexpressed as the claim that

(3.4) \(\displaystyle \ \ \ \ \ \ \ \displaystyle \lim_{ t \to 0 } \frac{ \mid f(p + tv) - f(p) - t df_p(v) \mid }{t} = 0\) ... ... "

Can someone please explain/demonstrate (formally and rigorously) exactly why/how the definition

\(\displaystyle df_p(v) = \displaystyle \lim_{ t \to 0 } \frac{ f(p + tv) - f(p) }{t}\)

implies that

\(\displaystyle \displaystyle \lim_{ t \to 0 } \frac{ \mid f(p + tv) - f(p) - t df_p(v) \mid }{t} = 0\) ... ... ?

**Question 2**In the above text from Tapp we read the following:

" ... ... \(\displaystyle \displaystyle \lim_{ t \to 0 } \frac{ \mid f(p + tv) - f(p) - t df_p(v) \mid }{t} = 0 \)

which really just says that

(3.5) \(\displaystyle \ \ \ \ \ \ \ \ f(p + tv) - f(p) \approx t df_p(v)\) ... ...

Can someone please explain/demonstrate (formally and rigorously) exactly why/how

\(\displaystyle \displaystyle \lim_{ t \to 0 } \frac{ \mid f(p + tv) - f(p) - t df_p(v) \mid }{t} = 0\)

means the same as

\(\displaystyle f(p + tv) - f(p) \approx t df_p(v)\) ... ...?

Hope that someone can help ...

Peter