lim_n→->∞infty(1/3+n^1000/2^n)^n calculate

djo57bgfrqn 2022-10-15 Answered
lim n ( 1 3 + n 1000 2 n ) n calculate
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Answers (1)

spornya1
Answered 2022-10-16 Author has 18 answers
If f ( n ) = ( 1 3 + n 1000 2 n ) n ,
g ( n ) = ln f ( n ) = n ln ( 1 3 + n 1000 2 n ) = n ln ( 1 3 ( 1 + 3 n 1000 2 n ) ) = n ( ln ( 1 / 3 ) + n ln ( 1 + 3 n 1000 2 n ) = n ln ( 3 ) + n ln ( 1 + 3 n 1000 2 n )
Since 3 n 1000 2 n 0 and ln ( 1 + x ) x for small x, n ln ( 1 + 3 n 1000 2 n ) n 3 n 1000 2 n 0 so g ( n ) n ln ( 3 ) = ln ( 3 n ) and f ( n ) 1 / 3 n 0.
Used ln, not exp.
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