# Why are these definitions of directional derivative equivalent? (By limit and by scalar product)

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.

• If it is Fréchet differentiability, then why is it $t$ and not $||h||=||tv||=|t|||v||=|t|$ in the denominator? – creepyrodent Oct 31 '18 at 18:45