#readwise
# Partial Fractions - Limit Method
## Metadata
- Author: [[brilliant.org]]
- Full Title: Partial Fractions - Limit Method
- URL: https://brilliant.org/wiki/partial-fractions-limit-method/
## Summary
Partial fraction decomposition simplifies a rational function into smaller parts. The limit method helps find the coefficients in this decomposition by examining what happens as the denominator approaches zero. Although it is not the most efficient method, it provides a solid mathematical foundation. In the example given, the partial fraction decomposition of $\frac{1}{x^2 + 4x - 5}$ results in $\frac{1}{6(x-1)} - \frac{1}{6(x+5)}.$
## Highlights
The limit method uses [limits](https://brilliant.org/wiki/epsilon-delta-definition-of-a-limit/) 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. ([View Highlight](https://read.readwise.io/read/01jftdcrb955n21d4nk06ymfed)) ^6fifss
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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}.
$
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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.
$\lim_{x \rightarrow 1}\left( \frac{A}{x-1}+\frac{B}{x+5} \right) = \lim_{x \rightarrow 1} \frac{A}{x-1}$
$\lim_{x \rightarrow 1}\frac{1}{x^2 + 4x - 5} = \lim_{x \rightarrow 1} \frac{A}{x-1}.$
Multiply both sides of this equation by the $(x-1)$ factor, then evaluate the limit: ^l9q2gi
$
\begin {aligned}
\lim_{x \rightarrow 1} \frac 1 {x+5} &= \lim_{x \rightarrow 1} A \\
\frac 1 6 &= A.
\end {aligned}
$
^9g7ukv
This same process is used to compute $B$. This time, the limit is taken as $x$ approaches $-5$:
$
\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
$
Thus, the partial fraction decomposition is:
$
\frac{1}{x^2 + 4x - 5} = \frac{1}{6(x-1)} - \frac{1}{6(x+5)}.
$
([View Highlight](https://read.readwise.io/read/01jftdcza2ng99ghpe4xd5b6ky))
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