The concept of 'Limit' is fundamental to the study of Calculus which is an important branch of advanced Mathematics. The precise definition of 'Limit' is difficult to understand without looking at the Geometrical interpretation of 'Limit' and some examples.
Geometrical Interpretation of Limit : When a regular polygon of 'n' sides is inscribed in a circle it appears that as 'n' increases infinitely there is possibility of the polygon coinciding with the circle. But theoretically for any large value of 'n' the polygon never coincides with the circle. In such a case, we say that the limiting shape of the polygon is circle as 'n' increases infinitely and by notation it is denoted as      
   (Regular polygon of 'n' sides inscribed in a circle) = circle.
  Limit of a function: Consider the function  

 Now as x approaches the value '2', values of the function are given in the form of a table.
     Here f(x) is not defined at x=2 but we observe that as x approaches to '2' the value of f(x) approaches to '4'. In this case we say that limiting value of f(x) is '4' as x approaches to '2'.
This is denoted as

Mathematical definition of Limit
    If 'f' is a function of a variable 'x' and defined in a deleted neighbourhood of 'a', the number 'l ' is said to be a limit of f(x) as x→a, if for any arbitrarily chosen positive number δ, however small it may be, there exists a corresponding number ε greater than zero such that |f(x) -1| < ε for all values of 'x', for which 0 < | x-a | < δ. It is denoted by

    In other words, a function f(x) is said to tend to a limit 'l ' as 'x' tends to 'a', if the difference between f(x) and l can be made as small as we please by taking x sufficiently nearer to a.
                            We write it as

Note: i) The function need not be defined at x = a.
ii) The value of f(x) at x = a has no effect on the existence or non existence of the limit.
Infinite Limit: Let f be a function defined in a deleted neighbourhood of 'a'. If for every K > 0 (however large), there exists a, δ > 0 such that that f(x) > K whenever 0 < |x-a| < δ and is written as  

Indeterminate Forms: While finding the limiting values of certain functions, forms such as may be encountered. 

Since for the same form in different cases, different limiting values may be obtained, these forms are called indeterminate forms.
     Methods like substitution, rationalisation, formula application may be employed to eliminate these forms.