Q:

How can one point determine a unique straight line in differentiation?

I found a similar question and a beautiful answer here. However I'm not able to fully understand the answer and have a question on the selected answer at:

Consider all the lines going through point $(x_0,f(x_0))$. For every line, the relative error should approach $0$ as $x$ approaches $0$ because all these lines go through point $(x_0,f(x_0))$, the linear approximation equals the function at $x=x_0$. What is special about the tangent line in relation to the relative error? Why does Arturo say only the tangent line makes the relative error zero ?

A:

(A bit too long for a comment)
Why only consider straight lines passing through $(x_0,f(x_0))$?  Just consider any (continuous) function $g$ passing through $(x_0,f(x_0))$.  Let $f$ and $g$ be defined on the same open set $A$.
We may define for any $x_0\in A$:
$$\text{$g$ touches $f$ in $x_0$} \iff
\lim_{x\to x_0}\frac{|f(x)-g(x)|}{|x-x_0|}=0.$$
It is easy to show that amongst all functions $g$ that touch $f$ in $x_0$ there is at most one of the form
$$x\mapsto f(x_0)+m(x-x_0),\text{ where $m$ is a linear function,}$$
which is called the derivative of $f$ in $x_0$.

