Calculus used in the sequel, is given below

Calculus is a branch of mathematics which deals with the study of the changes that takes place in the values of a function as the points in the domain changes. The idea of limit is the very beginning of calculus and we need to take some examples to have a clear idea of it. Let us consider a function f(x) = x. If we take x to be very close to zero, then f(x) also takes values very close to zero. It can be written as   In a more general manner, let f(x) = x-xo then, . If , then . Here l is called the limit of the function f(x). The symbolic representation of this is  We say is the expected value of f(x) at x = a given the values of f(x) near to the left of a. This value is called the left-hand limit of f(x) at x = a. We say  is the expected value of f(x) at x = a given the values of f near to the right of a. This value is called the right-hand limit of f(x) at a. If there is such a value for which the left-hand limit and the right-hand limit coincides, then we call that value as the limit of f(x) at x = a and denote it as . Algebra of limits: If f(x) and h(x) be two functions such that both and  exist. Then   For all real numbers of ?,   , here  If f(x) is a polynomial function, then  exists, which is given by,  Let f(x) =ao+a1x+a2x2+………..anxn . Therefore,  Or,  Or,   Or,  An important limit, which is very useful and used in the sequel, is given below as  We know,  Therefore,    For example  Or,  Or,  Here the point to remember is that this expression is valid for any rational number and a is positive. For the evaluation of the limits of trigonometric functions, we shall make use of the following limits which are given by    Let us take an example, . We know,    The following theorems are very helpful in solving limits of trigonometric functions. Theorem 1:  Let f(x) and h(x) be two real valued functions with the same domain such that f (x) ? h(x) for all x in the domain of definition, For some a, and both exists then    Theorem 2:  Let f(x), g(x) and h(x) be three real function such that f(x) ? g(x) ? h(x) for all x values. Then for some real number ?, if  , then,