Partial Fractions - Limit Method
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.
Limit Method
Find the partial fraction decomposition of the rational expression,
\[\frac{1}{x^2+4x-5}.\]
The denominator can be factored as \((x-1)(x+5).\) This gives the partial fraction decomposition form,
\[\frac{1}{x^2+4x-5}=\frac{A}{x-1}+\frac{B}{x+5}.\]
Observe what happens when we take the limit of the right-hand side as \(x\) approaches \(1.\) 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.
\[\begin{align} \lim_{x \rightarrow 1}\left( \frac{A}{x-1}+\frac{B}{x+5} \right) &= \lim_{x \rightarrow 1} \frac{A}{x-1} \\ \\ \lim_{x \rightarrow 1}\frac{1}{x^2+4x-5} &= \lim_{x \rightarrow 1} \frac{A}{x-1}. \end{align}\]
Multiply both sides of this equation by the \((x-1)\) factor, then evaluate the limit:
\[\begin{align} \lim_{x \rightarrow 1}\frac{1}{x+5} &= \lim_{x \rightarrow 1} A \\ \\ \frac{1}{6} &= A. \end{align}\]
This same process is used to compute \(B.\) This time, the limit is taken as \(x\) approaches \(-5:\)
\[\begin{align} \lim_{x \rightarrow -5}\left( \frac{A}{x-1}+\frac{B}{x+5} \right) &= \lim_{x \rightarrow -5} \frac{B}{x+5} \\ \\ \lim_{x \rightarrow -5}\frac{1}{x^2+4x-5} &= \lim_{x \rightarrow -5} \frac{B}{x+5} \\ \\ \lim_{x \rightarrow -5}\frac{1}{x-1} &= \lim_{x \rightarrow -5} B \\ \\ -\frac{1}{6} &= B \end{align}\]
Thus, the partial fraction decomposition is:
\[\frac{1}{x^2+4x-5}=\frac{1}{6(x-1)}-\frac{1}{6(x+5)}.\]