Yes, calculus allows us to understand and analyze the behavior of functions more precisely than using infinitesimals alone. Here are a few reasons why:

Rigor and precision: The limit-based approach in calculus is more rigorous and precise than the use of infinitesimals. This is because limits are defined in a precise and formal way, and they allow us to analyze functions in a more systematic and rigorous way.

Conceptual clarity: The use of limits in calculus provides a clear conceptual understanding of what is happening when we analyze the behavior of functions. It allows us to understand how functions behave as they approach certain values or as their inputs increase or decrease.

Connections to other areas of mathematics: The limit-based approach in calculus is connected to other areas of mathematics, such as real analysis, topology, and complex analysis. This makes it easier to apply calculus concepts to other areas of mathematics and to understand how different mathematical concepts are related.

Generalization to higher dimensions: The limit-based approach in calculus can be extended to functions of multiple variables, which is important in fields such as physics and engineering. This allows us to study the behavior of functions in higher dimensions and to apply calculus concepts to more complex systems.

Overall, while the use of infinitesimals was an important development in the history of calculus, the limit-based approach is now the standard way of studying the behavior of functions. It allows us to analyze functions in a more precise and rigorous way, and it has led to many important advances in mathematics and its applications.
spam fails_task lang_mismatch pii not_appropriate hate_speech sexual_content quality toxicity humor helpfulness creativity violence