What is the correct solution for the limit lim x → ∞ x sin ⁡ 1 x ?

Question

What is the correct solution for the limit       lim          x      →      ∞        x  sin  ⁡      1    x  ?

Turynka2f · Accepted Answer

lim          x      →      ∞        x  sin  ⁡      1    x    =      lim          x      →      ∞                  sin      ⁡              (                  1          x                )                    1      x        =      lim          u      →      0                  sin      ⁡      (      u      )        u    =  1

Garrett Black · Accepted Answer

The first line of reasoning is clearly incorrect: let f(x)=x,   g  (  x  )  =  (      x    2    +  1      )          −      1      . Then      lim          x      →      ∞        f  (  x  )  =  ∞and   g  (  x  )  ∈  (  0  ,  1  ] is clearly bounded on       R  . But      lim          x      →      ∞        f  (  x  )  g  (  x  )  =      lim          x      →      ∞            1          x      +      1              /            x        =  0.

What is the correct solution for the limit <munder> <mo movablelimits="true" form="prefix">

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