Compute lim x → 0 sin 3 ⁡ x ( 2 x ) 3

Question

Compute       lim          x      →      0                          sin        3            ⁡      x              (      2      x              )        3

Wade Atkinson · Accepted Answer

depending on the fact of       lim          x      →      0                  sin      ⁡      x        x    =  1            sin      3              8              x        3              =      1    8              sin      ⁡      x        x              sin      ⁡      x        x              sin      ⁡      x        x

ziphumulegn · Accepted Answer

Recall that if f is continuous, then      lim          x      →      a        f  (  g  (  x  )  )  =  f      (          lim              x        →        a              g    (    x    )    )    .Since the cube function   x  ↦      x    3   is continuous, we have                              lim                      x            →            0                                                              sin              3                        ⁡            x                                (            2            x                          )              3                                                          =                              (                          1              2                                      lim                              x                →                0                                                                    sin                ⁡                x                            x                        )                    3                                                  =                              (                          1              2                        )                    3                                                  =                  1          8                .

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