TITLE: Taking "Zooming in on a point of a graph" seriously
QUESTION [65 upvotes]: In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope.  This explanation never really translates into a formal definition - we instead approximate the tangent line by secant lines.
I seem to have found a way to obtain tangent lines (and more) by taking "zooming in" seriously.  
Example 1
Take the curve $y = x(x-1)(x+1)$.  
I want to find an equation for the tangent line to this curve at the origin.  So I zoom in on the origin with a microscope of magnification power $c$ (i.e. I stretch both vertically and horizontally by a factor of $c$) to obtain
$\frac{y}{c} = \frac{x}{c}(\frac{x}{c} - 1)(\frac{x}{c}+1)$.
Multiplying through by $c$ I have
$y = x(\frac{x}{c} - 1)(\frac{x}{c}+1) $
Now letting my magnification power go to infinity I have
$y = -x$
Which is the correct answer.
Example 2
Take the curve $y = x^2$.  
I want to find an equation for the tangent line to this curve at the point (3,9).  I first rewrite the equation as
$(y-9) + 9= ((x-3) + 3)^2$
so that I am focusing on the appropriate point.  To zoom on this point with magnification $c$ I have
$\frac{y-9}{c} + 9 = (\frac{x-3}{c} + 3)^2$.
$\frac{y-9}{c} + 9 = \frac{(x-3)^2}{c^2} + 6\frac{x-3}{c} + 9 $
Multiplying through by $c$ I have
$y - 9 = \frac{(x-3)^2}{c} + 6(x-3) $
Now letting my magnification power $c$ go to infinity I have
$y - 9 = 6(x-3)$
Which is the correct answer.
Example 3
Here is the example which actually motivated me to consider this at all:
Take the curve $y^2 = x^2(1 - x)$.
This is a cubic curve with a singularity at the origin, and so it doesn't really have a well defined tangent line.  It sort of looks like it should have two tangent lines (y = x, and y = -x), but it is a little bit tricky to formalize this.  Let's see what "zooming in" does:
$\frac{y^2}{c^2} = \frac{x^2}{c^2}(1 - \frac{x}{c})$
$y^2 = x^2(1 - \frac{x}{c})$
Letting $c$ go to infinity I have
$y^2 = x^2$, or $(y-x)(y+x) = 0$, which is the pair of lines I desired.
My Questions

Do any books take this approach when developing the derivative?
I would imagine that algebraic geometers do this kind of thing formally.  Is there a more rigorous analogue of the prestidigitation I engage in above?  Where would I look to read up on such things?

p.s.  It would be nice to illustrate each of these examples with a little movie of the "zooming in" process, but I am not sure how to put such things on MO. Any hints?

REPLY [5 votes]: Excellent question!  One of the earliest advocates of the "zooming in" approach is the mathematician and education specialist David Tall of Warwick; see e.g. http://www.tallfamily.co.uk/david/papers/2.embodied-calculus
His latest related paper is a reevaluation of Cauchy's legacy here.  The basic idea is that ideas of zooming and informal infinitesimals should be introduced before any formalisation takes place.  Once the student has a grasp of the key concepts, the course can branch out either into standard or non-standard calculus.