A Taylor series is a series expansion of a function about a point. A one-dimensional  Taylor series is an expansion of a real  function  about a point  is given by

 
1) Evaluating definite integrals
Here is an example. Suppose we want to evaluate the definite integral

2)Understanding asymptotic behaviour
 Taylor series can tell us useful information about how a function behaves in an important part of its domain. Here is an example 

A famous fact from electricity and magnetism says that a charge q generates an electric field whose strength is inversely proportional to the square of the distance from the charge. That is, at a distance r away from the charge, the electric field is
E = kq / r2      where k is some constant of proportionality.

3)Understanding the growth of functions

4) Solving differential equations
 
 
						

							
If a function  has continuous derivatives up to (n+1)th order, then this function can be expanded in the following fashion:



where , called the remainder after n+1 terms, is given by:

When this expansion converges over a certain range of , that is, , then the expansion is called the Taylor Series of  expanded about .
 
you can use taylor series to find out the limit of various expressions.consider the expression as f(x) in the given definition and write x as x+h where h tends to zero.generate the taylor series or expand it and put h=0 finally after all simplifications and the result will be the limit of the expression under consideration.
but always be very careful towards the definitions and conditions for existence of the taylor expansion of any expression.!!!!!!!!