In elementary textbooks on differential calculus, the notion of derivative number at, say, $x=a$ (i.e. $f'(a)$) is sometimes introduced via a parallelism between the analytic process and the geometric process, saying: while the difference quotient ($ \frac { f(a+h) - f(a)} { h} $) goes to a determinate limit (as $h$ goes to $0$), the straight line passing through $(a, f(a))$ and $(a+h, f(a+h))$ tends to a " limiting position", namely the tangent line to the graph of function $ f$ at point $(a, f(a))$.
Of course, this approach is intuitive, and using a graphing calculator we "see" that the line defined by $ y-f(a) = f'(a) ( x-a)$ is tangent to the graph of function $f$.
Here we seem to face an alternative: (1) either the tangent is defined as the line satisfying the above equation ( and in that case my question is pointless), or (2) we have a concept of tangent line that is independent from differential calculus , in such a way that the claim "the line passing through $(a, f(a))$ and having a slope equal to $f'(a)$ is tangent to the graph of $f$" has a substantial content.
In short: is it possible to prove, as an informative (and not simply definitional) claim that the line defined by $ y-f(a) = f'(a) (x-a)$ is the tangent to the graph of $f$ passing through $(a, f(a)$? And if so, which concept of "tangent " can be used to this effect?
