Illustration 8 Consider the function ( ) · Part 2

the function at x = is well defined and is, indeed, equal to , but the limit of the function at x = is not even defined. Illustration As a final illustration, we find f x , where f x ≠  =   Table . . . . . . . f ( x ) . . . . . . As usual we tabulate the values of f ( x ) for x near . From the values of f ( x ) for x less than , it seems that the function should take value at x = ., i.e., f x Similarly, the value of f ( x ) should be as dic- tated by values of f ( x ) at x greater than . i.e. f x But then the left and right hand limits coincide and hence f x f x f x Graph of function given in Fig . strengthens our deduction about the limit. Here, we Fig . Fig . MATHEMATICS note that in general, at a given point the value of the function and its limit may be different (even when both are defined). . . Algebra of limits In the above illustrations, we have observed that the limiting process respects addition, subtraction, multiplication and division as long as the limits and functions under consideration are well defined. This is not a coincidence. In fact, below we formalise these as a theorem without proof. Theorem Let f and g be two functions such that both lim → f ( x ) and lim → g ( x ) exist. Then (i) Limit of sum of two functions is sum of the limits of the functions, i.e., → [ f ( x ) + g ( x )] = lim → f ( x ) + lim → g ( x ) . Limit of difference of two functions is difference of the limits of the functions, i.e., → [ f ( x ) – g ( x )] = lim → f (