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I want to prove that the definitions of directional derivative by scalar product, $\nabla f(x) .v$, and by a limit, $\displaystyle \lim_{h \to 0}\dfrac{f(x+hv)-f(x)}{h}$, are the same one. I tried to do it using definitions of gradient but failed. Wikipedia proved it this way, but I can't understand the start of the proof, because I don't see why you can start from that limit being $0$. Isn't this an assumption of the thesis?

Note: While I was writing this post, I found an answer on why they are equivalent here, but I still want an explanation of what Wikipedia is doing, so I'll make the post.

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This topic is a minefield when it comes to notation inconsistencies if you're not careful.

The reason why Wikipedia set the answer of the limit of the last equation to 0 is because it uses the definition of Fréchet differentiability, which you can verify by clicking the blue differentiable link in your image.

In the implication part, the first equality is due to the fact that the gradient of a scalar function creates an R^n vector at any given x, which is a linear map satisfying the definition of Fréchet differentiability. The second equality comes straight from the algebra from above.

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  • $\begingroup$ If it is Fréchet differentiability, then why is it $t$ and not $||h||=||tv||=|t|||v||=|t|$ in the denominator? $\endgroup$ – creepyrodent Oct 31 '18 at 18:45
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    $\begingroup$ Fréchet differentiability implies directional derivative. $\endgroup$ – Trung Oct 31 '18 at 19:16

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