Nahgrenzwert

Nahgrenzwert, also known as the limit of a function, is a fundamental concept in calculus and mathematical analysis. It describes the behavior of a function as its input approaches a certain value. The limit of a function f(x) as x approaches a is denoted by the expression lim(x→a) f(x). This concept is crucial for understanding continuity, derivatives, and integrals.

The limit of a function exists if, for every positive number ε, there exists a positive number δ such that the absolute difference between f(x) and the limit L is less than ε whenever the absolute difference between x and a is less than δ. This formal definition ensures that the function's output gets arbitrarily close to L as the input gets sufficiently close to a.

There are several important properties of limits, including the sum, difference, product, and quotient of limits, as well as the limit of a constant times a function. These properties allow for the calculation of limits of more complex functions by breaking them down into simpler components.

Limits are also used to define continuity. A function is continuous at a point if the limit of the function as x approaches a equals the function's value at a. This concept is essential for understanding the behavior of functions and is widely used in various fields of mathematics and science.