Evaluating lim_(n->oo)(n(ln(n+2)- ln n)) But I can't figure out any good way to solve this. Is there a special theorem or method to solve such limits?

Chaim Ferguson

Chaim Ferguson

Answered question

2022-10-28

n ( log ( n + 2 ) log ( n ) ) = n n n + 2 d x x = 0 2 n n + x d x
so, by the dominated convergence theorem,
lim n + n ( log ( n + 2 ) log ( n ) ) = 0 2 1 d x = 2 .

Answer & Explanation

indivisast7

indivisast7

Beginner2022-10-29Added 13 answers

Why not elementary? n ( ln ( n + 2 ) ln n ) = ln ( n + 2 n ) n = ln ( 1 + 2 n ) n ln e 2 = 2
propappeale00

propappeale00

Beginner2022-10-30Added 5 answers

n ( ln ( n + 2 ) ln ( n ) ) = ln ( 1 + 2 / n ) 1 / n so the limit is 2 since it is the derivative of ln ( 1 + 2 x ) at 0.

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