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.

Find the partial fraction decomposition of the rational expression,

\[\frac{1}{x^2+4x-5}.\]

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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:

\[\frac{1}{x^2+4x-5}=\frac{1}{6(x-1)}-\frac{1}{6(x+5)}.\]