Who can help me solving this limit? lim x → 0 x 3 6 + sin ⁡ x − x x 5 I don't need answer Taylor Maclaurin L'Hospital and Derivatives. But I know my answer 1 120 How can I do?

Question

Who can help me solving this limit?      lim          x      →      0                                    x          3                6            +      sin      ⁡      x      −      x              x      5      I don&#039;t need answer Taylor Maclaurin L&#039;Hospital and Derivatives. But I know my answer       1    120  How can I do?

Malcolm Flores · Accepted Answer

Well you can use L&#039;Hospital rule 5 times because each time we get 0/0 form:      lim          x      →      0                  cos      ⁡      (      x      )              5      !        =      1          5      !

Ryland Houston · Accepted Answer

L        =                  lim                      x            →            0                                                                              x                3                            6                        +            sin            ⁡            x            −            x                                x            5                                              =                  lim                      x            →            0                                                                              x                3                            6                        +            3            sin            ⁡            (            x                          /                        3            )            −            4                          sin              3                        ⁡            (            x                          /                        3            )            −            x                                x            5                                                                  =                  lim                      x            →            0                                                              x              3                                      /                        6            −            3            (            x                          /                        3                          )              3                                      /                        6            −            4                          sin              3                        ⁡            (            x                          /                        3            )                                x            5                                                                  +                  lim                      x            →            0                                                              3              (              x                              /                            3                              )                3                                            /                            6              +              3              sin              ⁡              (              x                              /                            3              )              −              3              (              x                              /                            3              )                                                      3                5                            (              x                              /                            3                              )                5                                                                                            =                              L                          3              4                                      +                  lim                      x            →            0                                                4            (            x                          /                        3                          )              3                        −            4                          sin              3                        ⁡            (            x                          /                        3            )                                x            5                                                                  =                              L                          3              4                                                                              +                              4                          3              5                                                lim                      x            →            0                                                (            (            x                          /                        3            )            −            sin            ⁡            (            x                          /                        3            )            )            (            (            x                          /                        3                          )              2                        +                          sin              2                        ⁡            (            x                          /                        3            )            +            (            x                          /                        3            )            sin            ⁡            (            x                          /                        3            )            )                                (            x                          /                        3                          )              5                                                                              =                              L                          3              4                                      +                              2                          3              5                                            ∴  L  =            1      120

Who can help me solving this limit? lim_(x-> 0 )((x^3)/(6) +sin x - x )/(x^5)

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