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1

Solved example of Limits by factoring

$\lim_{x\to5}\left(\frac{x^2+8x-9}{x^3-x^2+5x-5}\right)$
2

We can factor the polynomial $x^3-x^2+5x-5$ using synthetic division (Ruffini's rule). We search for a root in the factors of the constant term $-5$ and we found that $1$ is a root of the polynomial

$1^3-11^2+5\cdot 1-5=0$
3

Let's divide the polynomial by $x-1$ using synthetic division. First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $1$ and multiply by the factor $1$. Add the result to the second coefficient and then multiply this by $1$ and so on

$\left|\begin{array}{c}1 & -1 & 5 & -5 \\ & 1 & 0 & 5 \\ 1 & 0 & 5 & 0\end{array}\right|1$
4

In the last row of the division appear the new coefficients, with remainder equals zero. Now, rewrite the polynomial (a degree less) with the new coefficients, and multiplied by the factor $x-1$

$\lim_{x\to5}\left(\frac{x^2+8x-9}{\left(5+x^{2}\right)\left(x-1\right)}\right)$
5

Factor the trinomial $x^2+8x-9$ finding two numbers that multiply to form $-9$ and added form $8$

$\begin{matrix}\left(-1\right)\left(9\right)=-9\\ \left(-1\right)+\left(9\right)=8\end{matrix}$
6

Thus

$\lim_{x\to5}\left(\frac{\left(x-1\right)\left(x+9\right)}{\left(5+x^{2}\right)\left(x-1\right)}\right)$
7

Simplifying

$\lim_{x\to5}\left(\frac{x+9}{5+x^{2}}\right)$
8

Evaluating the limit when $x$ tends to $5$

$\frac{5+9}{5+5^{2}}$
9

Simplifying

$\frac{7}{15}$