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Let g: R -> R be a differentiable function. Given that lim_(n->inf) n*(g(2009 + 1/n) - g(2009)) = 27 and lim_(alpha -> pi/2) g(2009 + cos(alpha)) = 3. Find lim_(t->0) (g(2009) / g(2009 + t))^(1/t).

1. e³
2. e²
3. e⁻²
4. e⁻⁹

Correct answer: e⁻⁹

Solution

g'(2009) = 27 (from definition of derivative) and g(2009) = 3. The limit lim_(t->0) (g(2009)/g(2009+t))^(1/t) = exp(lim_(t->0) ln(g(2009)/g(2009+t))/t) = exp(-g'(2009)/g(2009)) = exp(-27/3) = e⁻⁹.

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