The Limit of a Function

This is because the notion of a derivative is dependent on the existence of limits in the theory of functions. This is why the theory of limits of functions is considered to be the cornerstone of calculus. What exactly is a limit? To further grasp the notion of “limit,” consider the following example: consider the function f (x). Make it so that the independent variable x takes values that are close to a given constant a. Then, the function f(x) takes a set of values corresponding to x. Assume that when the value of x is close to ‘a,’ the values of f(x) are close to a particular constant. For example, let us suppose that by selecting values of x that are sufficiently close to a but not equal to a, we may have f(x) diverge arbitrarily slightly from the value ‘a’, and that this is true for all such values of x. Then, when x approaches ‘a’, it is argued that f(x) is approaching limit A. 

Limits and functions: 

A function may be on the verge of approaching two separate limitations. Two scenarios are possible: one in which the variable approaches its limit by values more than the limit, and another in which the variable approaches its limit through values less than the limit. In such a circumstance, the limit is not defined, but the right-hand and left-hand limits are both present and functional.

The right-hand limit of a function is the value that the function approaches when a variable approaches its limit from the right side of the function’s graph.

This can be expressed in the following way: 

limx→a f(x) = A+ 

The left-hand limit of a function is the value that the function approaches when a variable approaches its limit from the left side of the function’s graph.

This can be expressed in the following way: 

limx→a f(x) = A- 

History: 

Despite being implicit in the development of calculus during the 17th and 18th centuries, the modern concept of the limit of a function can be traced back to Bolzano, who in 1817 introduced the fundamentals of the epsilon-delta technique to define continuous functions in order to define continuous functions. His work, on the other hand, was not well-known during his lifetime.

Cahuy discussed variable quantities, infinitesimals, and limits in his 1821 book Cours d’analyse, and defined continuity of y=f(x)y=f(x) by stating that an infinitesimal change in x necessarily produces an infinitesimal change in y, while (Grabiner 1983) claims that he used a rigorous epsilon-delta definition in his proofs. In 1861, Weierstrass published the first version of the epsilon-delta definition of limit in the form that is still often used today. He also pioneered the use of the notations lim and limx→x0, among others. 

Properties: 

A function f is real-valued if and only if the right-handed limit and the left-handed limit of the function f at p are both equal to L. If the limit of the function f at p is real-valued, the limit of the function f at p is L.

F(x) approaches p and is equal to f(p), then the function is continuous at p, and else it is discontinuous (p). This is equivalent to saying that f transforms every sequence in M which converges towards p into a sequence in N which converges toward f. This is equivalent to saying that every sequence in M converges towards f(p).

Suppose N is an ordered vector space. The limit operation is linear in the following sense: if the limit of f(x) as x approaches p is L and the limit of g(x) as x approaches p is P, then the limit of f(x) + g(x) as x approaches p is L + P. 

Limit of a function of two variables: 

If we have a function f(x,y) which depends on two variables x and y. Then this given function has the limit say C as (x,y) → (a,b) provided that ϵ>0,∃ δ > 0 such that |f(x,y)−C| < ϵ whenever 0 < √(x-a)²+(y-b)² < δ.

It is defined as 

lim(x,y)→(a,b) f(x,y) = C 

Limits of functions and continuity: 

The concepts of limits and continuity are strongly tied to one another. Functions may be either continuous or discontinuous in nature. The continuity of a function is defined as follows: if there are minor changes in the input of the function, then there must also be small changes in the output of the function.

In elementary calculus, the condition f(X) -> λ as x -> a denotes that the number f(x) can be made to lie as close as we want to the number lambda as long as we take the number x to be unequal to the number a but close enough to a to be considered equal to a. This demonstrates that f(a) may be a long way from lambda and that it is not necessary to define f(a). The following is an extremely significant result that we utilise in the derivation of functions: f’(a) of a given function f at a number a can be thought of as, 

f’(a) = limx→a f(x) – f(a)/x-a 

Conclusion: 

The application of limits in calculus is not limited to the computation of derivatives and integrals; it has a wide range of applications in a variety of other domains as well. The following are examples of how Limits can be used: It is useful in determining the strength of magnetic fields, electric fields, and other fields; Limits are used to extract the most relevant pieces of information from huge complex functions that include a lot of information.