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

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djo57bgfrqn

Answered question

2022-10-15

lim n ( 1 3 + n 1000 2 n ) n calculate

Answer & Explanation

spornya1

spornya1

Beginner2022-10-16Added 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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