I'm studying differentiation in multiple variables, and I don't understand one expression.
Given a vector $u$, we fix the restriction of $f$ which goes trough the point $a$ and has direction $u$. In parametric form it is written $a+tu, t\in \mathbb{R}$ and see how this function changes in $t=0$, (the parameter that corresponds to points $a$ $$\lim_{t\to0} \frac{f(a+tu)-f(a)}{t}=D_f(a)$$ If this limit exist we call it the derivative of $f$ in direction $u$.
Now what I don't understand:
We can also write $f(a+tu)=f(a)+tD_uf(a)+o(t),t\rightarrow{0}$
What does $o(t)$ mean? I think I can understand $f(a+tu)$ geometrically as the images of a line generated by the values of $t$ and direction $u$. However with $f(a)+tD_uf(a)+o(t), t\rightarrow{0}$ I just don't see it.
It is a vector map such that $lim_{t\rightarrow 0}{{o(t)}\over t}=0$, you have $f(a+tu)-f(a)-tDf_a(u)=o(t)$.