Actual Infinity in Reality?

[polymath257]

How would you describe the limit as X goes to zero?

I suspect that when you include this, we are not that far apart in our thinking about the topic.  That this allows you to end the infinite regression (or progression) and reach the destination.

The limit as x goes to zero only exists if two other limits exist: Limit as x goes to 0+ and limit as x goes to 0-; these refer to the side from which zero is approached.
If these two limits are the same, the function is said to be continuous. If they are not, there is a discontinuity.
If both limits are +∞ and/or -∞, then you have a vertical asymptote.

Much like the concept of infinity (∞), the infinitesimal dx is also just a conceptual aid, it's not a number.

A couple of comments:

1. Infinitesimals are no longer used. They were problematic in several ways and were replaced by the epsilon-delta definition of limit, which does not require them.

2. For a function to be continuous, the limit from the left and the limit from the right have to be equal *and* that limit has to be the value of the function. The two one-sided limits being equal just means the limit exists. Continuity means that limit is also the value of the function (obtained by plugging in).