Please help me solve this question:

lim (x^3-3x-2)/

x->2 (x^2-5x+6)

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- Oct 7th 2007, 07:57 AMredpanda11limited thoughts on limits :S
Please help me solve this question:

lim (x^3-3x-2)/

x->2 (x^2-5x+6) - Oct 7th 2007, 08:02 AMJhevon
- Oct 7th 2007, 08:06 AMtopsquark
$\displaystyle \lim_{x \to 2} \frac{x^3 - 3x - 2}{x^2 - 5x + 6}$

Note that $\displaystyle x - 2$ is a factor of $\displaystyle x^3 - 3x - 2$, evidenced by the fact that $\displaystyle 2^3 - 3 \cdot 2 - 2 = 0$.

Thus, by long or synthetic division we get that

$\displaystyle x^3 - 3x - 2 = (x - 2)(x + 1)^2$

The denominator factors.

Thus

$\displaystyle \lim_{x \to 2} \frac{x^3 - 3x - 2}{x^2 - 5x + 6} = \lim_{x \to 2} \frac{(x - 2)(x + 1)^2}{(x - 2)(x - 3)}$

$\displaystyle = \lim_{x \to 2} \frac{(x + 1)^2}{x - 3}$

$\displaystyle = \frac{9}{-1} = -9$

-Dan