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?

Elementary Calculus;his treatment of the microscope methodology is based on work of Keith Stroyan. $\endgroup$ – Robert Haraway Oct 4 '11 at 21:11