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 )

Question

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    )

Jamarcus Shields · Accepted Answer

The pointwise limit f of the sequence   {      f    n        }          n      ∈                        Z                +             you get by computing, for every   x  &amp;gt;  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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