How to find the limit of lim n → ∞ ( n sin ⁡ n π 2 cos ⁡ 1 n )

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How to find the limit of       lim          n      →      ∞        (  n  sin  ⁡            n      π        2    cos  ⁡      1    n    )

Punktatsp · Accepted Answer

The basic strategy for showing that limits of this kind do not exist is to approach infinity along a different path; usually we consider what happens as n approaches   ∞ from the evens and the odds. I&#039;ve provided an example of how to do this with your problem as provided below.First note that as   n  →  ∞,   1      /    n  →  0 so   cos  ⁡  (  1      /    n  )  →  1; moreover, since   cos  ⁡  (  1      /    n  )  =  cos  ⁡  (  1  )  &amp;gt;  cos  ⁡  (  π      /    2  )  =  0 we have that   cos  ⁡  (  1      /    n  )  &amp;gt;  0 for every   n  ≥  1. Now note that  sin  ⁡  (  n  π      /    2  )  =      {                            ±          1                          n          ≡          1          ,          3                    mod                              4                                      0                          n          ≡          0          ,          2                    mod                              4                        so if we take   n  →  ∞ with n=2m then  2  m  sin  ⁡      (                  2        m        π            2        )    cos  ⁡      (          1              2        m              )    =  2  m  cos  ⁡      (          1              2        m              )    sin  ⁡  (  m  π  )  =  0while if we take   n  →  ∞ with n odd, then      |    n    sin    ⁡          (                        n          π                2            )        cos    ⁡          (              1        n            )        |    =      |    (    2    k    +    1    )    sin    ⁡          (                        (          2          k          +          1          )          π                2            )        cos    ⁡          (              1                  2          k          +          1                    )        |    =      |    (    2    k    +    1    )    cos    ⁡          (              1                  2          k          +          1                    )        |  which evidently approaches   ∞ as   n  →  ∞. Since we have approached   ∞ along two different paths (namely n even and n odd) that produce different values, we conclude that the limit does not exist

How to find the limit of <munder> <mo movablelimits="true" form="prefix">lim <mrow cla

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