classical mechanics - How to interpret instantaneous velocity using limit?

Mathematics is the language of physics therefore to understand physics is to understand its language. Based on definition of derivative the smaller the change of X of difference quotient the more it approach a certain value if it exist which we call the derivative. But what does it mean in physics? If derivative is the model use to describe instantaneous velocity then it means that the smaller you observe time interval experimentally, the ratio of distance over time approaches to a certain value. But why does it approach a certain value when we observe an object moves ? What causes it to approach a constant speed in smaller interval? and why call it the instantaneous velocity/speed? The concept of derivative only describe it but have no explanation.