Use the limit rule to find lim n → ∞ n 2 ( 1 − cos ⁡ ( 1 n ) )

Question

Use the limit rule to find       lim          n      →      ∞            n          2        (  1  −  cos  ⁡  (      1    n    )  )

Sydnee Villegas · Accepted Answer

Multiply top and (missing) bottom by   1  +  cos  ⁡  (  1      /    n  ). After using the identity   1  −      cos    2    ⁡  t  =      sin    2    ⁡  t, we get that we want      lim          n      →      ∞            (          1              1        +        cos        ⁡        (        1                  /                n        )              ⋅                            sin          2                ⁡        (        1                  /                n        )                    (        1                  /                n                  )          2                      )    .The rest should follow from a limit you know.

Feinsn · Accepted Answer

Start with the identity   1  −  cos  ⁡  (  2  x  )  =  2      sin    2    ⁡  (  x  ) to get                                          n            2                                    (                      1            −            cos            ⁡                          (                              1                n                            )                                )                                    =                              2                          sin              2                        ⁡                          (                              1                                  2                  n                                            )                                                          1                              n                2                                                                        (1)                                  =                  1          2                                      (                                          sin                ⁡                                  (                                      1                                          2                      n                                                        )                                                            1                                  2                  n                                                      )                    2                    Then use the limit                    (2)                              lim                      x            →            0                                                sin            ⁡            (            x            )                    x                =        1            to get                    (3)                              lim                      n            →            ∞                                    n          2                          (          1          −          cos          ⁡                      (                          1              n                        )                    )                =                  1          2

Use the limit rule to find <munder> <mo movablelimits="true" form="prefix">lim <mrow c

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