Why is lim n → ∞ n arctan ⁡ ( x n ) = x ?

Question

Why is       lim          n      →      ∞        n  arctan  ⁡  (      x    n    )  =  x  ?

vikafa4g · Accepted Answer

L&#039;Hôpital is overkill in this case. Put   h  =  x      /    n. Then as   x  →  ∞,   h  →  0, and  n  arctan  ⁡  (  x      /    n  )  =  x            arctan      ⁡      h        h    =  x            arctan      ⁡      h      −      arctan      ⁡      0        h    ,,and the limit of the fraction as   h  →  0 is by definition the derivative of arctanx at x=0

Monfredo0n · Accepted Answer

Try l&#039;Hospital:      lim          t      →      ∞        t  arctan  ⁡      x    t    =      lim          t      →      ∞                  arctan      ⁡              x        t                    1      t                          =                    l&#039;H                  lim          t      →      ∞                  −              x                  t          2                            1                  1          +                                    x              2                                      t              2                                                  −              1                  t          2                      =      lim          t      →      ∞          x            t      2                      x        2            +              t        2              =  xand thus the limit is not zero but x

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