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The function is $\"\\lim_{x\\rightarrow2}\\frac{x^2-2x}{x^2-x-2}\"$ .

Rewrite the function.

$\"=\\lim_{x\\rightarrow2}\\frac{x(x-2)}{(x+1)(x-2)}\"$

Cancel the common terms.

$\"=\\lim_{x\\rightarrow2}\\frac{x}{(x+1)}\"$

$\"\\\\=\\frac{2}{(2+1)}\\\\$

$\"\\lim_{x\\rightarrow2}\\frac{x^2-2x}{x^2-x-2}=\\frac{2}{3}\"$ .

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The function is $\"f(x)=\\frac{x^2-2x}{x^2-x-2}\"$ .

Construct the table for different value of $\"x\"$ , to estimate the value of $\"f(x)\"$ .

$\"x\"$	$\"f(x)\"$
$\"2.5\"$	$\"0.714286\"$
$\"2.1\"$	$\"0.677419\"$
$\"2.05\"$	$\"0.672131\"$
$\"2.01\"$	$\"0.667774\"$
$\"2.005\"$	$\"0.667221\"$
$\"2.001\"$	$\"0.666778\"$
$\"1.9\"$	$\"0.655172\"$
$\"1.95\"$	$\"0.661017\"$
$\"1.99\"$	$\"0.665551\"$
$\"1.995\"$	$\"0.606111\"$
$\"1.999\"$	$\"0.666556\"$

Observe the table.

The value of $\"f(x)\"$ at $\"x=1.999\"$ is $\"0.666556\"$ and the value of $\"f(x)\"$ at $\"x=2.001\"$ is $\"0.666778\"$ .

Therefore the value of $\"f(x)\"$ at $\"x=2\"$ is $\"0.666667\"$ .

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$\"\\lim_{x\\rightarrow2}\\frac{x^2-2x}{x^2-x-2}=\\frac{2}{3}\"$ .