Tangent line to a graph and derivatives? My book states:

Assuming that $f(x)$ is differentiable at $a$, the tangent line to the graph $y = f(x)$ at the point $(a, f(a))$ is given by the equation.
  $$f(x) - f(a) = f'(a)(x - a)$$

Now let me ask my question.
$f'(a)$ is an instant change (when $\Delta x \rightarrow 0$) of the function $f(x)$. If such, I can't get why we multiplying it by the $(x - a)$? I assume you can take any $x$ and any $a$. Thus the distance between them might be quite big. Why the product of non-infinite-small distance and derivative is valid? What am I missing there? 
The only way I could agree on that, is when
$$f(x) - f(a) = f'(a)(x - a)$$
$$x \rightarrow a$$
Then the distance $(x - a)$ is infinitely small and thus derivative is applicable.

The tangent line is a good approximation locally but it need not be a good approximation when $|x-a|$ is huge. 
To study the error term, If $f$ is of class $C^{2}$, then for each $x$, there exists a $c$ between $a$ and $x$ such that 
$$f(x)-p(x)=\frac{f^{(2)}(c)}{2}(x-a)^2$$
here $p(x)$ is the tangent line.
