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Limits of a function in Calculus


In Calculus, limit of a function is an important concept and it is use to know the behavior of the function near any definite input. Limits have many applications in the mathematics. A limit tells how a function comes close to a certain value, and is used to get the value of that function.

Let’s see the general notation of limits of Functions.

: lim x→c f(x) =L,

Where, L is the limit of function f(x) as function value (x) approaches other constant value c. When value of function x is closed to constant value c then function f(x) becomes close to limit of L. If there is no other value for this function then function is within same condition, like notation given below:

lim x→c f(x) =L,

Or f(x) → L as x → c,

The definition of a limits is not concerned with value of function of f(x) when function value x is equal to constant value c. So all we care about the values of the function f(x), when x is close to c, on either the left side or right side .

Limit Functions are also following the constant rule which is useful for working with and computing limits in this rule function f(x) is constant for all function value x then the limits as x approaches constant value c must be equal to constant value b. It means limit of function is

lim x→c b =b.

Definition of Limits of a function (Definite form)

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Limits of a function, can be defined as the approaching value of the function near a given input. Let’s take a function 'f' which, has the output f(x) to every input 'x' and the function has limit 'L' and an input 'p'. f(x) is close to 'L' , when f(x) is close to 'p'.

In simple words, we can say as f(x) moves closer to 'L' , 'x' moves closer to 'p'. Limits of a function, also describes its behavior as the independent variable increase or decrease without any restriction. In actual the value we get from the limit is not the exact value but, the value we get is very close to real value or we can say tends to that value

Limit of a function evaluation ( indeterminate form )

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Whenever we deal with limits we have to compare Numbers and Functions and after that we need the knowledge of limits and its indeterminate forms. In limits the indeterminate form is defined as an algebraic expression. Limits Indeterminate forms, means any form which can't be precisely determined or which doesn't possess a definite solution. The Limits indeterminate forms includes various Functions or numbers like:

00 , 0/0 ,∞ *∞, +∞/+∞ and many other such type of functions.
Its very easy to check the addition of two large functions as it can be written as: ∞+∞=∞ but, it’s impossible to know the division of large number (∞/∞), and it is called indeterminate form. May be one of the most important form is the division of 2 small numbers in which denominator is 0 it is also defined as an indeterminate form. While a number is close to +∞ we call it a large number and when a number is close to '- ∞' it is also called as a large number but, in negative direction. Many students have confusion that '-∞' is a very small number but, in reality, it a is big number but in negative direction. Always remember that, the inverse of large number is a small number and inverse of small number is a large number. Like:
1/∞=0, in this we divide the 1 by ∞, which will give 0 as result.
1/0=∞, On the other side, if we divide the 1 by 0 result will be ∞.
This is all about Limits indeterminate form.