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Evaluate lim(x to oo) sqrt(x) (sqrt(x -...

Evaluate `lim_(x to oo) sqrt(x) (sqrt(x - 2007) - sqrt(x))`

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We have
`underset(x to oo)(lim) sqrt(x) (sqrt(x - 2007) - sqrt(x))`
`= underset(x to oo)(lim) sqrt(x) ((sqrt(x - 2007) - sqrt(x)))/(sqrt(x - 2007) + sqrt(x)) (sqrt(x - 2007) + sqrt(x))`
`= underset(x to oo)(lim) sqrt(x) (x - 2007 - x)/(sqrt(x - 2007) + sqrt(x)) ((oo)/(oo) "form")`
`underset(x to 00)(lim) (-2007)/(sqrt(1 - (2007)/(x)) + 1)`
`= (-2007)/(2)`
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