Calculate: lim x → 0 + tan ⁡ x ⋅ tan ⁡ x − sin ⁡ x ⋅ sin ⁡ x x 3 x

Question

Calculate:      lim          x      →              0        +                        tan      ⁡      x      ⋅              tan        ⁡        x            −      sin      ⁡      x      ⋅              sin        ⁡        x                            x        3                    x

Donavan Mack · Accepted Answer

You can first remove a few factors      lim          x      →              0        +                        tan      ⁡      x      ⋅              tan        ⁡        x            −      sin      ⁡      x      ⋅              sin        ⁡        x                            x        3                    x                =      lim          x      →              0        +                        tan      ⁡      x      ⋅              tan        ⁡        x                    x              x                  lim          x      →              0        +                        1      −      cos      ⁡      x              cos        ⁡        x                    x      2          =      lim          x      →              0        +                        1      −      cos      ⁡      x              cos        ⁡        x                    x      2        .Then multiply by the conjugate  =      lim          x      →              0        +                        1      −              cos        3            ⁡      x                      x        2            (      1      +      cos      ⁡      x              cos        ⁡        x            )        ,evaluate the finite factor at denominator  =      1    2        lim          x      →              0        +                        1      −      cos      ⁡      x      (      1      −              sin        2            ⁡      x      )              x      2        ,use trigonometric identitites  =      1    2        lim          x      →              0        +                        2              sin        2            ⁡              x        2            +      cos      ⁡      x              sin        2            ⁡      x              x      2        ,and conclude  =            (              1        2            )        2    +      1    2    .We used            tan      ⁡      x        x    =            sin      ⁡      x        x        1          cos      ⁡      x        →  1  ,            sin      ⁡      a      x        x    =  a            sin      ⁡      a      x              a      x        →  a  .

Finley Mckinney · Accepted Answer

lim                      x            →                          0                              +                                                                          tan            ⁡            x            ⋅                          tan              ⁡              x                        −            sin            ⁡            x            ⋅                          sin              ⁡              x                                                          x              3                                      x                                                          =                  lim                      x            →                          0                              +                                                                                        tan                              3                                  /                                2                                      ⁡            x            −                          sin                              3                                  /                                2                                      ⁡            x                                              x              3                                      x                                                                        =                  lim                      x            →                          0                              +                                                                                        sin                              3                                  /                                2                                      ⁡            x                                x                          3                              /                            2                                                                          sec                              3                                  /                                2                                      ⁡            x            −            1                                x            2                                                            =                  lim                      x            →                          0                              +                                                                          (                                          sin                ⁡                x                            x                        )                                3                          /                        2                          ⋅                  lim                      n            →            ∞                                                              sec                              3                                  /                                2                                      ⁡            x            −            1                                x            2                                                            =                  lim                      x            →                          0                              +                                                                                        sec                              3                                  /                                2                                      ⁡            x            −            1                                x            2                                                            =                  lim                      x            →                          0                              +                                                                                                        (                1                +                                  1                  2                                                  x                  2                                +                                  1                  24                                                  x                  4                                +                ⋯                )                                            3                                  /                                2                                      −            1                                x            2                                                            =                  lim                      x            →                          0                              +                                                                                        (              1              +                              3                4                                            x                2                            +              ⋯              )                        −            1                                x            2                                                            =                  3          4

Calculate: <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXA

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