Basic Concepts of Limits:

1. Meaning of the Phrase “Tend to Zero”:
            Suppose a variable 'x' assumes in succession a set of values.
                                                               
Clearly, 'x' is becoming smaller and smaller as n is increasing and can be made as small as we want by taking 'n' sufficiently large. This un-ending decrease of 'x' is symbolically expressed by “” and read as “x tends to zero”.
It should be noted that the symbol “” is quite different from. The equationmeans that 'x' has actually assumed the value zero whileimplies that the variable 'x' takes in succession a serried of values which becomes smaller and smaller such that the difference between x and 0 becomes and remains less than any pre-assigned positive number, however small.
2. Meaning of the Phrase “x Tends to Infinity”:
            Let a variable 'x' assumes the values
                                                               
It is evident from the above values that 'x' will get larger and larger values as 'n' increases and can be made 'n' sufficiently large.
This unending increase of 'x' is symbolically expressed by “” and is read as “x tends to infinity”. This simply indicates endless progress of x to numerically greatness without band.
3. Meaning of the Phrase “x Tends to a”:
            Suppose 'x' assumes in succession a set of values.
                                                               
Clearly, the successive difference of these values from 'a' are
                                                               
Which become smaller and smaller as 'n' increases and can be made small as we like by taking 'n' sufficiently large.
This behavior of 'x' is symbolically expressed by “” implies that the variable x takes in succession a set of values that approaches nearer and nearer to 'a' in such a manner that the numerical value of remain less than any pre-assigned positive number, however small.
Geometric Illustration of a Limit:
            Let OAB be a circle and let OB be a chord intersecting the circumference at O and B. Suppose the chord OB is to rotate in a clockwise direction about O, then the point of intersection B will move along the circumference, the arc OB and chord OB decrease.
                        
Let the rotation continue until B is infinitely close to 'O' and the chord and the arc become infinitely small. It can be conceived that in the limiting position whose B moves to coincide with O, i.e. the two points on intersection coincide — the straight line does not cut the circumference in the second part. Therefore, in the limiting position, the chord becomes a tangent to the circle at O.
Limit of a Function:
            If the value of the functionapproaches a fixed number L as 'x' approaches 'a', we say that L is the limit of functionand 'x' approaches 'a'.
Symbolically
                                                               
                        We read as
            “Limit ofis L as x approaches a”.
Left and Right Limit:

Left Hand Limit:
            Let 'a' be any fixed point. If 'x' approaches to 'a' through the value of 'x' less than 'a'. Then the limit is called left hand limit.
Thus, limit is devoted by
                                                               
Right Hand Limit:
            Let 'a' be any fixed point. If 'x' approaches to 'a' through the values of 'x' greater than 'a'. Then the limit is called right hand limit.
Thus, limit is  denoted by
                                                               
Theorems of Limits:

• The limit of a function, if exist, is unique.
• 
• , where c is constant.
• 
• 
• 
• 
• 

Sandwich Theorem:
            Iffor all numbers 'x' in some interval containing a except possibly at a itself such that
                                                                , then
                                                                         

(Nature of Functions)

Function and Limits

(examples-of-limit)