Limits and Continuity

The concept of the limit is critical to grasp in order to prepare for calculus. A limit is a value that a function approaches as the value of its independent variable approaches a specified value. For instance, given the function f (x) = 3x, one could assert, “The limit of f (x) as x approaches 2 is six.” This is denoted symbolically by f (x) = 6. In the following sections, we will define a limit more precisely and provide instances of functional limitations to better illustrate the notion.

Another far-reaching term in calculus is continuity. A function’s nature may be continuous or discontinuous. One simple technique to determine a function’s continuity is to examine if the function’s graph can be traced with a pen without lifting the pen from the paper. A conceptual explanation of continuity such as this one is probably sufficient for the arithmetic we do in precalculus and calculus, but a more technical definition is required for higher math. By utilising limits, we’ll also discover a considerably more exact method of defining continuity. You are prepared for calculus if you comprehend the principles of limits and continuity.

Limits

Limits are described in mathematics as the values at which a function approaches its output for certain input values. Calculus and mathematical analysis make considerable use of limits to define integrals, derivatives, and continuity. It is employed during the analysis process and always refers to the function’s behaviour at a certain time. The concept of a sequence’s limit is further developed in the concept of a topological net’s limit, which is related to the concept of a sequence’s limit and direct limit in the theory category. Integrals are generally classified into two types: definite integrals and indefinite integrals. The upper and lower bounds of definite integrals are properly specified. Whereas indefinite integrals are expressed without bounds and use an arbitrarily large constant to integrate the function. Let us discuss in depth the definition and representation of the function’s limits, along with their attributes and examples.

What Are Limits?

In mathematics, limits are defined as unique real numbers. Consider a real-valued function “f” and a real number “c.” In general, the limit is defined as xc f(x) = L. It is read as “the maximum value of f for x as x approaches c equals L.” The “lim” denotes the limit, and the right arrow denotes the fact that function f(x) approaches the limit L as x approaches c.

Characteristics of Limit 

The following are some properties of the function’s limits: If the constraints limx→alimx→a f(x) and limx→alimx→a g(x) exist and n is a positive integer, then

• Addition Law: 

xa[f(x)+g(x)]=xaf(x)+xag(x)

• Subtraction Law: 

xa[f(x)−g(x)]=xaf(x)−xag(x)

• Multiplication Law:

xa[f(x)⋅g(x)]=xaf(x)⋅xag(x)

• Division Law: 

xa[f(x)g(x)] = xaf(x)xag(x),  where xag(x) ≠ 0 , xa[f(x)g(x)] = xaf(x) xag(x) , where xag(x) ≠ 0

• Constant Law:

xac=c

Continuity

The continuity and differentiability of a function are mutually exclusive. Before proving the function y = f(x) at the point x = a, it must first be proved for its continuity at the point x = a. Geometrically and algebraically, the concepts of continuity and differentiability can be established.

The continuity of a function f(x) at the point x = c can be established if the function’s limit at that point is equal to the function’s value at that point. xcf(x)=f(c). The derivative of a function y = f(x) is defined as f'(x) or d/dx.f(x) and is denoted by the expression f′(x)=xc(f(x+h)-f (x))/ h.

Continuity Of A Function

Continuity can be simply defined as the ability to draw a graph y = f(x) without lifting the pencil at a point. Let f(x) be a real-valued function on the subset of real numbers and c be a point included inside the domain of f(x). The function f(x) is said to be continuous at the position x = c if xcf(x)=f(c).

A function’s continuation can be illustrated graphically or algebraically. In a graph, the continuity of a function y = f(x) at a point is a graph line that travels through the point continuously and without interruption. The continuity of a function y = f(x) can be observed algebraically by comparing its value from the left-hand limit to its value from the right-hand limit.x1-f(x)=x1+f(x). That is, for values of x = 0.99, 0.998, which are little less than 1, the f(x) function has the same value as for values of x = 1.001, 1.0001, which are slightly larger than 1.

Conclusion

The concept of limits and continuity is one of the most critical concepts to grasp in order to perform calculus correctly. A limit is defined as the value that a function reaches when its independent variable reaches a specified value, and continuity is a critical feature of functions due to the way it interacts with other qualities. In elementary calculus, continuity of a function is a required but not sufficient condition for differentiation and integration.