Solve limit lim x → 0 x − sin ⁡ ( x ) ( x sin ⁡ ( x ) ) 3 / 2

Question

Solve limit       lim          x      →      0                  x      −      sin      ⁡      (      x      )              (      x      sin      ⁡      (      x      )              )                  3                      /                    2

Julien Carrillo · Accepted Answer

You have with Taylor series around 0 :  sin  ⁡  (  x  )  =  x  −            x      3        6    +  o  (      x    3    )      lim          x      →      0                  x      −      sin      ⁡      (      x      )              (      x      sin      ⁡      (      x      )              )                  3                      /                    2                      =      lim          x      →      0                  x      −      x      +                        x          3                6            +      o      (              x        3            )              (      x      (      x      +      o      (      x      )              )                  3                      /                    2                      =      lim          x      →      0                                    x          3                6            +      o      (              x        3            )                      x        3            +      o      (              x        3            )        =      1    6

Mackenzie Rios · Accepted Answer

For   x  →  0 we have   (  sin  ⁡  x      )          3              /            2        ∼      x          3              /            2       and so            x      −      sin      ⁡      x              (      x      sin      ⁡      x              )                  3                      /                    2                      ∼                    x        3                    3        !                            x                  3                      /                    2                    (      x              )                  3                      /                    2                      =      1    6

Solve limit <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXAt

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