Erroneously approximating a continuous object with a limit of a discrete one.

Question

Example: How does a planet going around the Sun? We discretize the movements to small periods when the planet flies forward and when it falls, then let $n\to\infty$ and we get the correct answer. But why? What is the reason behind?

Basically, I am looking for a similar example (maybe from real life), when we apply the same strategy, and we $do \ not$ get the correct answer. That would be very, very cool. That would show to apply caution with similar questions, and always argue why taking the limit is acceptable.

Answer 1

Black-body radiation is such an example that led Planck to initialize quantum theory.

Answer 2

Here's an easy one:

Let's say we want to figure out what $\sqrt{2}$ is, that is, the distance alone this line:

But we can only measure vertical/horizontal steps. So, we approximate it by discrete steps:

The total length of these discrete steps is 2. In fact, if we take the limit of these steps, and assume that the limit of the length is the length of limit:

We would come to the conclusion that $2 = \sqrt{2}$.

Of course, this isn't really the "real world" per se, but it's an example of discretization giving you the wrong answer.