How do I evaluate the limit lim x → 0 ⌊ tan ⁡ x sin ⁡ x x 2 ⌋

Question

How do I evaluate the limit      lim          x      →      0            ⌊                  tan        ⁡        x        sin        ⁡        x                    x        2              ⌋

Elijah Benjamin · Accepted Answer

Using Taylor–Young expansions: we know that  tan  ⁡  (  x  )  =  x  +            x      3        3    +  o  (      x    4    )  ,    sin  ⁡  (  x  )  =  x  −            x      3        6    +  o  (      x    4    )  ,hence  tan  ⁡  (  x  )  sin  ⁡  (  x  )  =      x    2    +            x      4        6    +  o  (      x    5    )  ,and hence            tan      ⁡      (      x      )      sin      ⁡      (      x      )              x      2        =  1  +            x      2        6    +  o  (      x    3    )  .From here, we conclude that there exists a punctured neighborhood V of 0 such that  ∀  x  ∈  V  ,               tan      ⁡      (      x      )      sin      ⁡      (      x      )              x      2        &amp;gt;  1.Since      lim          x      →      0                  tan      ⁡      (      x      )      sin      ⁡      (      x      )              x      2        =  1we can assume that V has been chosen such that  ∀  x  ∈  V  ,     1  &amp;lt;            tan      ⁡      (      x      )      sin      ⁡      (      x      )              x      2        &amp;lt;  2.Hence  ∀  x  ∈  V  ,         ⌊                  tan        ⁡        (        x        )        sin        ⁡        (        x        )                    x        2              ⌋    =  1  ,from which we conclude that      lim          x      →      0            ⌊                  tan        ⁡        (        x        )        sin        ⁡        (        x        )                    x        2              ⌋    =  1.

How do I evaluate the limit <munder> <mo movablelimits="true" form="prefix">lim <mrow cl

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