@Sudharshan: Great question. You’ll notice that the change between x and x^2 (2x + 1) is actually dependent on the size of the change you are measuring. If you are jumping from 2^2 to 3^2, you take a “step” of 1.0 and a change of 9 – 4 = 5 = 2x + 1.

What about taking a smaller step, such as 2^2 to 2.1^2? We’re only jumping to the number .1 in front of us.

In this case, the difference is 4.41 – 4 = .40 + .01 = 2*x*(.1) + (.1)^2

You’ll see that the “error term” is based on how far you step! In Calculus, we take tiny, microscopic steps which means the error term is some microscopic amount squared (micro-microscopic). For small steps, our error rate is shrinking faster than our step rate, and eventually becomes negligible.

The trick: to measure the difference from 2^2 to 3^2, don’t jump all at once. Find the difference from 2^2 to 2.1^2, and 2.1^2 to 2.2^2, and so on… the error at each stage is (.1)^2 = .01, so after 10 jumps the total error is only .1.

So, jumping from 2 to 3 in steps of .1 gives a total error of .1. If we jumped in steps of .000001, we’d have a total error of .000001. At some point, we can make the steps small enough to be “accurate enough” for our needs (there’s always some error threshold we can work within).

I plan on writing more about this!