Partial fraction decomposition is a method to express a rational function as a sum of simpler rational expressions. The limit method uses limits as a denominator factor approaches zero to compute the coefficients of the partial fraction. Although this method is less efficient than other partial fraction decomposition methods, it provides a mathematically rigorous basis for some of these more efficient methods.
Find the partial fraction decomposition of the rational expression,
The denominator can be factored as This gives the partial fraction decomposition form,
Observe what happens when we take the limit of the right-hand side as approaches The first rational expression will approach infinity, while the second rational expression will approach a constant. Therefore, the limit of this sum is equal to the limit of just the first rational expression.
Multiply both sides of this equation by the factor, then evaluate the limit:
This same process is used to compute This time, the limit is taken as approaches
Thus, the partial fraction decomposition is: