epsilondeltakriteerinä

The epsilon-delta criterion, often referred to as the definition of a limit, is a fundamental concept in mathematical analysis used to formally define the limit of a function. It provides a rigorous way to state that a function approaches a specific value as its input approaches a certain point. The criterion states that for a function f(x), the limit of f(x) as x approaches c is L, written as lim x→c f(x) = L, if and only if for every epsilon (ε > 0) there exists a delta (δ > 0) such that if 0 < |x - c| < δ, then |f(x) - L| < ε.

In simpler terms, the epsilon-delta criterion means that we can make the output of the function f(x) arbitrarily close to the limit L by choosing an input x sufficiently close to c. The epsilon (ε) represents a small positive value defining how close we want f(x) to be to L. The delta (δ) represents a corresponding small positive value that dictates how close x must be to c to achieve this closeness in the function's output. If we can find such a delta for any given epsilon, then the limit is formally established. This definition is crucial for proving theorems and understanding continuity in calculus and beyond.