I have to prove, without L'hopital rule, the following limit: lim x → ∞ x sin ⁡ 1 x = 0I tried doing a variable change, setting t = 1 x and reaching the following: lim t → 0 1 t sin ⁡ tBut I can't prove neither.

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I have to prove, without L&#039;hopital rule, the following limit:      lim          x      →      ∞            x    sin  ⁡      1    x    =  0I tried doing a variable change, setting   t  =      1    x   and reaching the following:      lim          t      →      0                  1      t        sin  ⁡  tBut I can&#039;t prove neither.

mnaonavl · Accepted Answer

See that       |    sin  ⁡  x      |    ≤  x while   x  →  0 and here             1      x        →  0 so  −            1      x        ≤  sin  ⁡            1      x        ≤            1      x      multiply RH and LH by       x   gives us  −      x              1      x        ≤      x    sin  ⁡            1      x        ≤      x              1      x      or  −            1              x              ≤      x    sin  ⁡            1      x        ≤            1              x            finally the sandwich theorem concludes that      lim          x      →      ∞            x    sin  ⁡            1      x        =  0

I have to prove, without L'hopital rule, the following limit: <munder> <mo movablelimits="

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