# Limit of a Function in Mathematics.

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## Limit of a Function in Mathematics

Introduction:

In calculus and mathematical analysis, limits are important because they help define concepts like integrals, derivatives, and continuity. It is employed in the analysis process and constantly considers how the function will behave at a specific point. In the third century BC, Archimedes of Syracuse devised the idea of limits to measure the volume of a sphere and curves.

The necessary quantity can be obtained by first dividing these numbers into small, approximatively pieces, then multiplying the number of parts. In the late 17th century, Gottfried Wilhelm Leibniz and Isaac Newton each simultaneously developed the infinitesimal calculus. These discoveries were later given a stronger conceptual foundation through work that included codifying the concept of limits. Calculus is now widely used in social science, engineering, and science.

Calculus and mathematical analysis depend on limits, which are also used to determine integrals, derivatives, and continuity. The concept of the limit of a topological net expands the definition of the limit of a sequence and relates it to the limit and direct limit in theory category. In general, there are two types of integrals: definite integrals and indefinite integrals.

Suppose we are watching the values of a function ƒ(x) as x approaches (without taking on the value of itself). Certainly, we want to be able to say that ƒ(x) stays within one-tenth of a unit from L as soon as x stays within some distance. But that in itself is not enough, because as x continues on its course toward what is to prevent ƒ(x) from jittering about within the interval from to without tending toward L- (1 / 10) to L+ (1 / 10)?

Each time, we find a new about so that keeping x within that interval satisfies the new error tolerance. And each time the possibility exists that ƒ(x) jitters away from L at some stage. The figures on the next page illustrate the problem. You can think of this as a quarrel between a skeptic and a scholar. The skeptic presents to prove that the limit does not exist or, more precisely, that there is room for doubt.

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The scholar answers every challenge with an around that keeps the function values within of L. How do we stop this seemingly endless series of challenges and responses? By providing that for every error tolerance that the challenger can produce, we can find, calculate, or This leads us to the precise definition of a limit

Definition:

Let f(x) be defined on an open interval x0 about except possibly at itself x0 .We say that limit of f(x) approaches is the number L and write

Lim x→ x0 f(x) = L

If for every number ε>0 there exist a corresponding number such that for all x,

0 < | x - x0| <

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