# Chain rule Calculator

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### Difficult Problems

1

Example

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

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

$\begin{matrix}\left(-1\right)\left(9\right)=-9\\ \left(-1\right)+\left(9\right)=8\end{matrix}$
3

Thus

$\lim_{x\to5}\left(\frac{\left(9+x\right)\left(x-1\right)}{-5+5x-x^2+x^3}\right)$
4

We can factor the polynomial $-5+5x-x^2+x^3$ 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

$-5+1\cdot 5+1^2\left(-1\right)+1^3=0$
5

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

$\left|\begin{array}{c}1 & -1 & 5 & -5 \\ & 1 & 0 & 5 \\ 1 & 0 & 5 & 0\end{array}\right|1$
6

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 $x-1$

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

Simplifying the fraction by $x-1$

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

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

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

Simplifying

$\frac{7}{15}$