What is the point of the remainder term in the Taylor series Take the first order taylor approximation:
We say $f$ is differentiable at $x_0 \in S^o$ if $\exists \space \nabla f(x_0)$ and a function $R: \mathbb{R}^n \to \mathbb{R}$ such that
$f(x) = f(x_0) + \langle \nabla f(x_0), x-x_0 \rangle + R(x-x_0)$
I don't understand what the point of the Remainder term is? Is it just trying to say there is a degree of error involved in the taylor approximation (which is the first two terms if I am not mistaken?)
What does this limit: $\lim_{x \to x_0} \frac{R(x-x_0)}{\| x-x_0 \|}=0$ convey in terms of information? If this implies that $R(x-x_0)$ grows slower than $\|x-x_0\|$ why is that useful?

The behavior of the error is the only thing that really lets us know that it's a good approximation.
I can, for example, write $x^3=5x\sin(x)+xe^x+R(x)$, where $R(x) = x^3 -5x\sin(x)-xe^x$, and call $R$ the error. Is this useful in any way?
