Manipulating lim x → 0 x + x x n For x → 0 determine the order of smallness relative to x of the function x + x I need to find n for lim x → 0 x + x x n ≠ 0Here's what I've got so far: lim x → 0 x + x x n = lim x → 0 x + x x 2 n ⟶ lim x → 0 x + x x 2 n = lim x → 0 x + x x 2 n x − x x − x = lim x → 0 x 2 − x x 2 n ( x − x ) = lim x → 0 x − 1 x 2 n − 1 ( x − x ) At this point, I'm not sure how to manipulate the fraction. Any hints?Once I finish manipulating the limit to the form lim x → 0 1 x f ( n ) , I can find n by solving f ( n ) = 0

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Manipulating       lim          x      →      0                          x        +                  x                            x        n            For   x  →  0 determine the order of smallness relative to   x of the function       x    +          x      I need to find   n for       lim          x      →      0                          x        +                  x                            x        n              ≠  0Here&#039;s what I&#039;ve got so far:      lim          x      →      0                          x        +                  x                            x        n              =      lim          x      →      0                                    x          +                      x                                    x                      2            n                                ⟶      lim          x      →      0                          x        +                  x                            x                  2          n                      =      lim          x      →      0                          x        +                  x                            x                  2          n                                    x        −                  x                            x        −                  x                      =      lim          x      →      0                                    x          2                −        x                              x                      2            n                          (        x        −                  x                )              =      lim          x      →      0                          x        −        1                              x                      2            n            −            1                          (        x        −                  x                )            At this point, I&#039;m not sure how to manipulate the fraction. Any hints?Once I finish manipulating the limit to the form       lim          x      →      0                  1              x                  f          (          n          )                    , I can find   n by solving   f  (  n  )  =  0

Alyvia Marks · Accepted Answer

If       lim          x      →      0                          x        +                  x                            x        n              =  c for some   c  ≠  0, then       lim          x      →      0                          x        +                  x                            x                  2          n                      =      c    2  . Now, let       x    =  t. We then wish to find   n such that      lim          t      →      0                          t        2            +      t              t              4        n              ≠  0It&#039;s easy to see that   n  =      1    4   will produce a limit of   1. Now, if   α  =  4  n  &amp;lt;  1, we may multiply the numerator and denominator by       t          −      α       to obtain that the limit equals      lim          t      →      0                t          2      −      a        +      t          1      −      α      and since each exponent above is positive, the limit is   0. It follows that   n  =      1    4   is the smallest   n for which the limit is nonzero.EDIT: Rewriting this answer in terms of ancient mathematician&#039;s comment is perhaps simpler. Letting   x  =      t    4   in the initial limit, we may rewrite it as  L  =      lim          t      →      0                          t        4            +              t        2                    t              4        n              =      lim          t      →      0                t          1      −      4      n        ⋅            t      2        +    1    ,and from the first term it&#039;s easy to see that   n  &amp;lt;      1    4   implies   L  =  0 while   n  =      1    4   implies   L  =  1

I need to find n for lim_(x->0)((sqrt{x+sqrt(x)))/(x^n))!=0

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