What is lim x → 4 x 4 - 256 x 3 - 64 ?

Question

What is             lim              x        →        4                                      x          4                -        256                              x          3                -        64            ?

kiriyama4b4 · Accepted Answer

Step 1We can discover: using polynomial long division or synthetic division:                              XXX                            (              x        4            -      256      )        =          (      x      -      4      )              (              x        3            +      4              x        2            +      16      x      +      64      )      and                              XXX                            (              x        3            -      64      )        =          (      x      -      4      )              (              x        2            +      4      x      +      16      )      Step 2As long as       x    ≠    4                                XXX                                              x          4                -        256                              x          3                -        64              =                            x          3                +        4                  x          2                +        16        x        +        64                              x          2                +        4        x        +        16                        lim              x        →        4                                      x          4                -        256                              x          3                -        64              =                                        (            4            )                    3                +        4                              (            4            )                    2                +        16                  (          4          )                +        64                                          (            4            )                    2                +        4                  (          4          )                +        16                                          XXXXXXXXX                      =                  4                  (          64          )                            3                  (          16          )                      =          16      3

diento7nq · Accepted Answer

lim              x        →        4                                      x          4                -        256                              x          3                -        64                  =          lim              x        →        4                                      x          4                -                  4          4                                      x          3                -                  4          3                    Multiplying both numerator and denominator by             (      x      -      4      )      I came up with this idea because there is a theorem in limits that goes as follows:            lim              x        →        a                                      x          n                -                  a          n                            x        -        a              =    n          a              n        -        1            Thus,      =          lim              x        →        4                                      x          4                -                  4          4                            x        -        4              ⋅                  x        -        4                              x          3                -                  4          3                          =          lim              x        →        4                                      x          4                -                  4          4                            x        -        4              ⋅          lim              x        →        4                            x        -        4                              x          3                -                  4          3                          =                  4        ⋅                  4          3                            3        ⋅                  4          2                          =                  4        2            3      Answer:       =          16      3

What is lim_(x rightarrow 4)(x^4-256)/(x^3-64)?

Answered question

Answer & Explanation

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