DEFINITION OF LIMIT

Despite being implicit in the development of calculus in the 17th and 18th centuries, the current concept of a function limit can be traced back to Bolzano, who established the epsilon-delta approach to define continuous functions in 1817. His work, however, was unknown during his lifetime. Hardy is responsible for the contemporary notation of placing the arrow below the limit symbol, which he developed in his 1908 book A Course of Pure Mathematics.

Limit

The value that the function approaches as its argument approaches a is the limit of a function at a point a in its domain (assuming one exists). In calculus and analysis, the concept of a limit is crucial. It’s used to determine derivatives and definite integrals, as well as to investigate the local behaviour of functions near places of interest. When a function has a limit, it is said to have a limit. If the function can be arbitrarily close to L by choosing values closer and closer to a, it is called L at a. It’s worth noting that the actual value at a has no bearing on the value of the limit.

A function may reach one of two limits. One in which variable approaches its limit by taking the values that are greater than the limit, and the other in which the variable approaches its limit by taking values that are smaller. The limit is not stated in this situation, although the right and left-hand limits do exist.

When a variable approaches its limit from the right, the right-hand limit of a function is the value of the function.

The result of a function approaches its left-hand limit when the variable approaches its limit from the left.

Existence of limit: –

For a limit to exist at any value of x, say x=c, it must be equal to the limit approaching from the right of c and the limit approaching from the left of c.

If LHL(Left Hand Limit)=RHL(Right-Hand Limit) then limit exists.

If the limit approaching from the right of c and the limit approaching from the left of c are not equal then the limit doesn’t exist at that point.

If LHL not equal to RHL then the limit doesn’t exist.

ALGEBRA OF LIMITS:-

The limits are assessed using algebraic methods. The factorization approach, evaluation using standard limits, direct substitution method, rationalisation, and evaluation of limits at infinity are all essential methods. When one of the numbers can be changed or at least one of the numbers is unknown, algebra is commonly used in formulas.

Let F(X) and G(X) be the two functions and their limit exists at a.

Then, 

1. The sum of the limits of two given functions is equal to the limit of the sum/subtraction of two functions.

• [{F(X)+G(X)}]=LimF(X) + LimG(X)

2. The difference of the limits of two given functions is equal to the limit of the subtraction of two functions.

• Lim[{F(X)-G(X)}]=LimF(X) – LimG(X)

3. The product of the limits of any two given functions is the same as the product of those function’s limits.

• Lim[{F(X)*G(X)}]=LimF(X) * LimG(X)

4. When we have the quotient of two functions, the limit of that term is the same as the quotient of their limits, with the exception that the denominator’s limit must not be 0.

• Lim[{F(X)/G(X)}]=LimF(X) / LimG(X) where G(x) not equals to 0.

5. A constant multiple of the function f(x) has a limit equal to c times the function’s limit.

• Lim(C*F(X))=C*Lim(F(X))

Limit of composition function:-

(f o g)(x) denotes the composition of two functions f(x) and g(x), implying that the range of the function g(x) should be inside the domain of the function f. (x). We now utilize the following property to calculate the limit of the composition of the two functions:

Lim fog(X)= Lim f((g(x))= f(Limg(x))

Limit tends to infinity:-

As x approaches infinity, the function f(x) has a real limit l if, no matter how small a distance we choose, f(x) moves closer to l and stays closer no matter how huge x becomes.

If we take a large number, the function f(x) tends to infinity as x tends to infinity.

f(x) grows to be larger than this number and stays that way no matter how big x grows to be.

If, however large and negative x is, the function f(x) tends to minus infinity as x tends to infinity, the function f(x) tends to minus infinity as x tends to infinity.

f(x) becomes more negative than the number we chose and remains more negative, no matter how big x becomes

CONCLUSION:-

Theory of the calculus is built on the concept of limits. Limits of functions are used to construct derivatives of functions, verify for function continuity, and so on. The value of a function’s limit at a certain moment offers us an intuitive impression of the function’s approaching value. It’s worth noting that when we calculate the limits, we’re not looking for the precise value of the function at that specific point. We’re more concerned with determining the direction or the point at which the function will occur. As x approaches a real number, the limit of f(x) approaches the value f(x) approaches as x approaches that real number.

We say that the limit of f(x) as x approaches c is L if the limit of f(x) as x approaches c is the same from both the right and the left.

We say the limit does not exist if f(x) never approaches a certain finite value as x approaches c. The two-sided limit does not exist if f(x) has the different right and left limits.