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logiski9s 2022-07-07 Answered
For all n N , let f n : ( 0 , ) R be the function defined by f n ( x ) = n sin ( x / n ) x ( x 2 + 1 ) . Find the pointwise limit of ( f n )
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Jamarcus Shields
Answered 2022-07-08 Author has 17 answers
The pointwise limit f of the sequence { f n } n Z + you get by computing, for every x > 0, the limit
f ( x ) = lim n f n ( x ) = lim n n sin ( x / n ) x ( x 2 + 1 ) .
Notice that (substituting h = 1 n ) this is the same limit as f ( x ) = lim h 0 sin ( x h ) x h 1 x 2 + 1 = 1 x 2 + 1 ,
and, so as you can see, we get the same limit that you computed.
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