Are Limits in Calculus Just Predictions or Mathematically Certain Truths?

If limits in calculus are about predicting the behavior of a function near a point, how can we be sure that the predicted value is mathematically true? Since predictions don’t always guarantee correctness, how does the formal definition of a limit (with epsilon and delta) ensure that the limit is not just a guess but a mathematically valid result? How do we verify that the limit truly represents the function's behavior?

Additionally, does this mean that calculus itself is built on a foundation of approximation rather than exactness? If so, how can we be sure that derivatives and integrals—both based on limits—yield completely accurate results rather than just close estimations?