Notion of limit

The notion of limit is a mathematical notion based on proximity. Limits are a method for analysing the inclination or behaviour of a function as the variable (input) nears a fixed value. We will learn about the notion of limit in this article in great detail. Practice without theory renders the theory useless. However, practice without an understanding of the theory and theoretical concepts can cause errors in your practice of limit questions. 

Before we learn about the notion of limit, let’s discuss function.

Function

A function is a method or procedure that connects each unique input with precisely one corresponding output. It remains at the centre of mathematics. Before calculus, students are taught that functions may be represented in a variety of forms, including formulae, tables, graphs and even words.

Functions are particularly important in calculus because they quite often prototype essential phenomena.  For example, the site of a moving object at a particular time, the rate at which a vehicle consumes gasoline at a set speed, a patient’s response to the magnitude of a drug dose. calculus can be used to investigate how output quantities respond differently to changes in the input variable. Furthermore, considering notions such as average and instantaneous velocity naturally takes us from an initial function to a related function, which is often the more intricate function.

Notion of limit

The original proposition of calculus, as used by Isaac Newton, Gottfried Leibnitz, Jacob and other greats, asserted that there can be infinitesimal quantities, that is, nonexistent small quantities, such that when an unlimited number of those quantities added together it would produce a finite number. While such a concept allowed the ancients to infer formulae for several traditional mathematical concepts, like the circumference and area of a circle, it was inadequately expressed, resulting in contradictions and philosophical debates.

We are adapted to the notion of a function y = f(x) as a formula that maps every real number described by x,  to only ONE real number y. In a mathematical analysation process, we care about the behaviour of functions within the points where they were described. 

For example, it might be easy to conclude that the functions y=x  and y=sinx  are equal to 0 at  x=0, but there is no clarity about the value of the following at x=0

 Both x  and sinx are 0 at   x=0, but we understand that we cannot operate on division, because division by 0 does not exist in maths. So, we need an approach, in which we see functions in the neighbourhood of the point that we are interested in. This approach is called the Theory of limits/notion of limit. 

Limits are a technique of studying a function’s tendency or pattern as the input variable nears a fixed value, or when the input variable rises or declines without bound.

Limit is a mathematical notion based on the concept of proximity that is used to give values to specific functions at locations in which no values are stated in a way that is compatible with neighbouring values. Since division by 0 is not a legitimate mathematical expression, the function (x2 – 1)/(x -1) isn’t defined when x is 1. The numerator may be factored & divided by (x – 1) to get x + 1 for any other given value of x.  As a result, for all values of x apart from 1, this quotient equals x + 1. Nevertheless, 2 could be ascribed to the function (x2 -1)/(x -1), as its limit, as x approaches 1 rather than its value when x = 1.

If there is a continuous function g(x) so that g(x) = f(x) throughout some interval surrounding x0, except maybe at x0 itself, then this is one approach of determining the limits of a function f(x) at the point x0. 

If, for almost any desired degree of closeness, one could identify an interval around x0 such that all values of f(x) determined here vary from L by an amount less than ε (i.e., if |x−x0| < δ, then |f (x)−L| < ε) then the following more fundamental explanation of limit, irrespective of the idea of continuity, may be given:

This concept may be used to assess if a particular number is a limit or not. Limits, particularly quotients, are frequently calculated by manipulating the function, so that it may be expressed in a way that makes the limit more evident, as in the example of (x2– 1)/(x – 1).

Conclusion 

Limits are the technique for calculating a function’s derivative/rate of change, and they’re utilised all across the analysis to turn approximations into accurate values, as when the area within a curved region is described as the limit of approximations by rectangles. In this article, we discussed elaborately the theory of limit/notion of limit. The theoretical understanding is crucial for practical applications. Read through the article thoroughly to gain a good grasp of the notion of limit.