unreasonably effective

The limit of a function at , , feels a lot like the value of the function . In fact, for simple functions like , , , the two are identical, they evaluate to the same number. Yet, the limit is different from a function’s value in general.

With limit, we do not bother looking at what the function is doing at . All that matter is what the function is doing in an (open) interval around . The interval can be as tiny as we wish, but cannot be a single point. In fact, even if the function is undefined at , the limit may still exist. Take for an example. At this one is undefined. But . For the Heaviside function, at , it is other way around; it has value but not limit. Both limit and function’s value may exist at , yet it is alright to have .

Without limit though, we would not have been able to talk about a function like speed, because the speed function (say ) is defined as the rate of change in the position function (say ). Without considering an interval (opposed to an instant) of time, it is hard to imagine what change would look like. And then the limit of the difference quotient involving is the “value” of the speed function: .

The difference between limit and function’s value may serve as a smoking gun to detect if a function is predictable, therefore if a function’s actual value is what it looks like from the point’s neighborhood.