The limit of a function is the most fundamental concept in the study of Calculus as well as of mathematical analysis.
Limit of a function at a point is defined by where is read as approaches (or tends) to and .
Consider a function defined by .
The domain of this function is the set of real numbers. .
It is natural to ask whether or not becomes closer and closer to as becomes closer and closer to but not equal to .
Hence i.e., as
The statement is also expressed by saying that limit of at is equal to
Algebra of Limits:
Let and be two functions such that both and exists , then
A point is at a distance less than from the point , then the set of all such that is at a distance less than any fixed from is expressed as
“ is arbitrarily close to and ”.
The neighborhood of a Point :
If , then any open interval containing the point is called a neighborhood of a point.
The open interval is called the – neighborhood of the point where .
Definition of Limit of Function:
A number is called the limit of a function as tends to i.e. if for any there exist depending on such that :
Let , we take then if taking we see that there exists depending in such that:
Left hand and Right-hand Limit of a Function
For any there exists depending on such that and , then
and means or
So if (i) and (ii)
Left hand limit
A number is called the left-hand limit of at or simply if for any there exists depending on such that
A number is called the right-hand limit of at it for any there exists depending in such that i.e.
i.e. , then exists and equal to
(ii) If , then does not exist.
(iii) For a function to have a limit at a point it is necessary and sufficient that both and exist, and these limits coincide.
(i) ,if given , there exists depending on such that .
(ii) , if given there exists depending on such that where is as large as possible number greater than .
Limits at Infinity:
(i) if given , there exists depending on such that
(ii) if given , there exists depending on such that
(iii) if given , there exists such that .The concepts and are also defined by infinite limits at infinity.
We choose , then if we take .
Thus given , there exists depending on such that .
If increases indefinitely through positive values, remains positive and decreases indefinitely. So intuitively we see that as .
and can be written as and in that case , holds if is replaced by and in second case is replaced by
(i) If a function satisfies the inequality in a neighborhood of and a function is such that , for and .
(ii) If in a neighborhood of the point and if is such that , for and then
Sandwich theorem / Squeezing theorem:
If and a function is such that for all in a deleted neighborhood of , then .
If , then find by sandwich theorem.
Since , with , we have and so by Sandwich Theorem: