Epsilondeltamääritelmät

Epsilondeltamääritelmät, often referred to as the epsilon-delta definition, are a rigorous way to define limits in calculus. Developed by mathematicians like Augustin-Louis Cauchy and Karl Weierstrass, these definitions provide a precise framework for understanding what it means for a function to approach a certain value as its input approaches another value.

The epsilon-delta definition of a limit states that the limit of a function f(x) as x approaches c is L, written as lim_{x→c} f(x) = L, if for every positive number ε (epsilon), there exists a positive number δ (delta) such that if 0 < |x - c| < δ, then |f(x) - L| < ε. In simpler terms, this means that we can make the output of the function, f(x), arbitrarily close to L by choosing an input, x, sufficiently close to c, but not equal to c. The ε represents the allowed "tolerance" for the output f(x) to be away from L, and the δ represents the corresponding "tolerance" for the input x to be away from c.

This definition is crucial because it avoids relying on intuitive notions of "closeness" and provides a purely logical and verifiable method for proving limit statements. It forms the foundation for many other concepts in calculus and real analysis, including continuity and derivatives. The epsilon-delta definition is applicable to both single-variable and multivariable calculus.