# Step-by-step Solution

## Find the limit of $\frac{x^2-5x+6}{3x^3-5x^2-x-2}$ as $x$ approaches $2$

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$-\frac{1}{15}$

## Step-by-step explanation

Problem to solve:

$\lim_{x\to2}\left(\frac{x^2-5x+6}{3x^3-5x^2-x-2}\right)$
1

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

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

Let's divide the polynomial by $x-2$ using synthetic division. First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $3$ and multiply by the factor $2$. Add the result to the second coefficient and then multiply this by $2$ and so on

$\left|\begin{array}{c}3 & -5 & -1 & -2 \\ & 6 & 2 & 2 \\ 3 & 1 & 1 & 0\end{array}\right|2$

$-\frac{1}{15}$
$\lim_{x\to2}\left(\frac{x^2-5x+6}{3x^3-5x^2-x-2}\right)$