epsilondeltamenetelmällä

Epsilondeltamenetelmä, often referred to as the epsilon-delta definition of a limit, is a rigorous way to define the concept of a limit in calculus. It provides a precise mathematical framework for understanding how a function behaves as its input approaches a certain value. The core idea is to show that for any arbitrarily small positive number, epsilon, we can find another positive number, delta, such that if the input is within delta of the target value (but not equal to it), then the output of the function is within epsilon of the limit value.

Formally, the statement "the limit of f(x) as x approaches c is L," written as lim_{x->c} f(x) = L, means that for every epsilon > 0, there exists a delta > 0 such that if 0 < |x - c| < delta, then |f(x) - L| < epsilon. The term |x - c| represents the distance between x and c, and |f(x) - L| represents the distance between f(x) and L. The epsilon-delta definition essentially states that we can make the output f(x) arbitrarily close to L by choosing an input x sufficiently close to c. This definition is fundamental to proving many theorems in calculus and real analysis.