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Jazmine Bruce 2022-05-18 Answered
Determine lim n arctan ( n + 1 ) arctan ( n )
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Answers (2)

Julien Carrillo
Answered 2022-05-19 Author has 13 answers
Note,
arctan n + 1 arctan n = arctan n + 1 n 1 + n + 1 n
= arctan 1 ( 1 + n 2 + n ) ( n + 1 + n )
Thus,
lim n arctan ( n + 1 ) arctan ( n )
= lim n arctan 1 ( 1 + n 2 + n ) ( n + 1 + n ) = arctan ( 0 ) = 0
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Nicholas Cruz
Answered 2022-05-20 Author has 3 answers
The arctan function is the inverse function of
tan : ( π 2 , π 2 ) R
as this function is monotonically increasing, we have
lim x π 2 tan x = + lim x + arctan x = π 2
therefore
lim n arctan ( n + 1 ) = π 2
lim n arctan ( n ) = π 2
so the difference is 0.
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