Title: Calculus: a limitless perspective Show Chinese title

Title: 微积分：无限的视角

Abstract: We propose a novel foundation for calculus that focuses on the notion of approximations while avoiding the use of limits altogether. Continuity is defined as approximation at a point, while differentiability is defined as approximation with a linear function. The errors in approximation are defined as a class of functions with certain properties; rules for combining error functions lead to all the familiar results in differential calculus. We believe that this approach is more natural for students while still giving a rigourous foundation to differential calculus. We demonstrate its utility by deriving the basic differential rules for trigonometric, hyperbolic and exponential functions, as well as L'H\^opital's Rule, Taylor polynomials, and the Fundamental Theorem of Calculus, all via approximation. Show Chinese abstract

Abstract: 我们提出了一种微积分的新基础，它专注于近似概念，同时完全避免使用极限。 连续性被定义为在一点处的近似，而可微性被定义为用线性函数进行的近似。 近似的误差被定义为具有某些性质的函数类；组合误差函数的规则导致微分学中的所有熟悉结果。 我们认为这种方法对学生来说更为自然，同时仍为微分学提供了严格的理论基础。 我们通过近似方法推导了三角函数、双曲函数和指数函数的基本微分法则，以及洛必达法则、泰勒多项式和微积分基本定理。