Calculus: Limit

Introduction

Limits are really simple and all of calculus is based on limits. If we define a function, say; f(x)= x-1/x-1, where the numerator and the denominator are the same and it can be simply written as 1, it will not be true. Rather it will be ‘almost’ true as there must be a constraint that ‘x’ must not be equal to 1 because, if the value of x is 1, then the function will become x(1) = 1-1/1-1 which will be 0 divided by 0 which will be undefined. 

Body

Calculus meaning 

Calculus can easily be stated as one of the core branches of mathematics. This branch of mathematics particularly deals with continuous change. The 2 basic concepts on which the whole theory of calculus is based are integrals and derivatives. In this context, the integral is defined as a method of uniting several parts so that the whole value can be calculated. Here, we come across a particular function differential that is often provided. Hence, integration or integral is the inverse function of differentiation. 

On the other hand, a function’s derivative is defined as the process of calculating the given function’s rate of change. The derivative of a function helps in explaining that function at a particular point whereas that function’s integral gathers its discrete values over the range of its values. While discussing calculus meaning it should also be stated that Calculus is often referred to by the name infinitesimal calculus. Infinitesimal numbers are described as those numbers that have quantities nearly equal to 0 but do not attain the value 0. In General terms, classical calculus involves the studying of continuous alterations in different provided functions. 

What is Calculus?

The question of what is calculus is quite important in mathematics. The branch of calculus focuses on some vital topics namely integration, differentiation, continuity, limit function, differentiability, and so on. Hence, Calculus is a particular section of mathematics that deals with the prospect of the rate change. This particular section of mathematics was developed by Leibniz and Newton. 

Normally calculus is utilised in different mathematical models for obtaining the best possible solutions. This helps in gaining a proper understanding of the alterations in different values that are related to a particular function. Calculus can be broadly categorised under two main sections. These are integral calculus and differential calculus. Both integral calculus and differential calculus serve as the basis for a higher mathematical branch. This is the mathematical branch of ‘Analysis’. This branch of mathematics particularly deals with the effects of a slight change in the variable that is dependent, as it ultimately leads to zero on the function. 

Limit calculus

In mathematics, limits are rather important. They are defined as the value which the particular function approaches the value of output for the provided input value. The concept of limits is very important in mathematical analysis as well as calculus and is utilised to define continuity, derivatives, and integrals. It is further utilised in the process of analysis and puts the main focus on a function’s behaviour at a particular point. A sequence’s limit is often generalised under the concept of a topological net’s limits and is considered concerning direct limit as well as limit under theory category. 

Normally integrals are of two types namely indefinite integral and definite integral and in the case of definite integral, the lower limits and upper limits are properly defined. This is where limit calculus comes under consideration. An indefinite integral becomes a definite integral when limits are applied to it. Consider the integration expression ∫cd f(y) dy for a particular function f(y) having limits [c, d] where d is considered as the upper limit and c is considered as the lower limit. To calculate the overall limit, firstly the integration of the given function is done which provides its antiderivative, after which the application of the limits are done on the calculated antiderivative. This has been outlined in the following. 

∫cd f(y) dy = [F (y)]dc = F(c) – F(d).

Conclusion

The overall article has been written on the core mathematical topic of Calculus: Limit. The topic is rather important in the subject of mathematical engineering. Limits in calculus, most of the time refer to the application of limits in the case of the definite integral. The core topic has been thoroughly analysed through discussion of several subtopics namely calculus meaning, what is calculus, and limits calculus.