How to evaluate this limit: lim x → 0 ( sin ⁡ x − x ) ( cos ⁡ x − 1 ) x ( e x − 1 ) 4

Question

How to evaluate this limit:      lim          x      →      0                          (        sin        ⁡        x        −        x        )        (        cos        ⁡        x        −        1        )                    x        (                  e          x                −        1                  )          4

benedictazk · Accepted Answer

lim                      x            →            0                                                              (              sin              ⁡              x              −              x              )              (              cos              ⁡              x              −              1              )                                      x              (                              e                x                            −              1                              )                4                                                                        =                  lim                      x            →            0                                                              (                                                −                                      x                    3                                                                    3                  !                                            +                                                x                  5                                                  5                  !                                            −              ⋯              )                                      (                                                −                                      x                    2                                                                    2                  !                                            +                                                x                  4                                                  4                  !                                            −              ⋯              )                                            x                                          (                x                +                                                      x                    2                                                        2                    !                                                  +                                                      x                    3                                                        3                    !                                                  +                ⋯                )                            4                                                                        =                  lim                      x            →            0                                                              x              3                                      (                                                −                  1                                                  3                  !                                            +                                                x                  2                                                  5                  !                                            −              ⋯              )                                      x              2                                      (                                                −                  1                                                  2                  !                                            +                                                x                  2                                                  4                  !                                            −              ⋯              )                                                          x              5                                                      (                1                +                                  x                                      2                    !                                                  +                                                      x                    2                                                        3                    !                                                  +                ⋯                )                            4                                                                        =                  lim                      x            →            0                                                              (                                                −                  1                                                  3                  !                                            +                                                x                  2                                                  5                  !                                            −              ⋯              )                                      (                                                −                  1                                                  2                  !                                            +                                                x                  2                                                  4                  !                                            −              ⋯              )                                                          (              1              +                              x                                  2                  !                                            +                                                x                  2                                                  3                  !                                            +              ⋯              )                        4                                                            =                  (                                    −              1                                      3              !                                )                          (                                    −              1                                      2              !                                )                                                  =                  1          12

Feinsn · Accepted Answer

Using basic limits                              lim                      x            →            0                                    (                                    sin              ⁡              x              −              x                                      x                              3                                              )                                    =                    −                  1          6                                              lim                      x            →            0                                    (                                    cos              ⁡              x              −              1                                      x                              2                                              )                                    =                    −                  1          2                                              lim                      x            →            0                                    (                      x                                          e                                  x                                            −              1                                )                                    =                    1            which can be evaluated using L&#039;HR, it follows that                              lim                      x            →            0                                                (            sin            ⁡            x            −            x            )            (            cos            ⁡            x            −            1            )                                x            (                          e                              x                                      −            1                          )                              4                                                                        =                              lim                      x            →            0                                    (                                    sin              ⁡              x              −              x                                      x                              3                                              )                          (                                    cos              ⁡              x              −              1                                      x                              2                                              )                                      (                          x                                                e                                      x                                                  −                1                                      )                                4                                                            =                              (                      lim                          x              →              0                                                          sin              ⁡              x              −              x                                      x                              3                                              )                          (                      lim                          x              →              0                                                          cos              ⁡              x              −              1                                      x                              2                                              )                                      (                          lim                              x                →                0                                                    x                                                e                                      x                                                  −                1                                      )                                4                                                            =                              (          −                      1            6                    )                          (          −                      1            2                    )                                      (            1            )                                4                                                            =                              1          12                .

How to evaluate this limit: <munder> <mo movablelimits="true" form="prefix">lim <mro

Answered question

Answer & Explanation

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