Let differentiable g : R → R satisfy lim α → π /2 g ( 2009 + cos α ) = 3 and lim n → ∞ n [ g ( 2009 + 1/ n ) − g ( 2009 )] = 27 . Find lim t → 0 + ( g ( 2009 + t ) g ( 2009 ) ) 1/ t .

Question

Let differentiable g : R → R satisfy lim α → π /2 ​ g ( 2009 + cos α ) = 3 and lim n → ∞ ​ n [ g ( 2009 + 1/ n ) − g ( 2009 )] = 27 . Find lim t → 0 + ​ ( g ( 2009 + t ) g ( 2009 ) ​ ) 1/ t .

TestHub · Accepted Answer

Continuity gives g ( 2009 ) = 3 , while the second limit gives g ′ ( 2009 ) = 27 . Taking logarithms yields − g ′ ( 2009 ) / g ( 2009 ) = -9 , so the limit is e − 9 .

Mathematics - Continuity - Differentiability Question with Solution | TestHub

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