On p. 22 of "Applied Differential Geometry" by William Burke he says that Df(u) must be the unique linear operator satisfying

$\displaystyle \lim_{h\rightarrow0}\frac{||f(u+h) - f(u) - Df(u)||}{||h||} = 0$

Why not just say:

$\displaystyle

Df(u) \equiv \lim_{h\rightarrow0}||\frac{f(u+h) - f(u)}{h}||

$

??

Then in the following example he says that Df(u) is a map.