Find the limit given that f ( 1 ) = 1, f ( x + y ) = f ( x ) + f ( y ) + 2 x y and f ( 1 x ) = f ( x ) x 4

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Find the limit given that   f  (  1  )  =  1,   f  (  x  +  y  )  =  f  (  x  )  +  f  (  y  )  +  2  x  y and   f      (          1      x        )    =            f      (      x      )              x      4

conveneau71 · Accepted Answer

Then the limit becomes  L  =      lim          x      →      0                          e                  −          2                      x            2                              −                        1          +          x                      ²                          3                    ln      ⁡      (      1      +              x        2            )        =      lim          x      →      0                  (      1      −      2              x        2            +      2              x        4            −      ⋯      +      ⋯      )      −      (      1      +                        x          2                3            −                        x          4                9            +      ⋯      −      ⋯      )                      x        2            −                        x          4                2            +                        x          6                3            +      ⋯      Simplifying this gives  L  =      lim          x      →      0                  −                        7                      x            2                          3            +                        19                      x            4                          9            −      ⋯      +      ⋯                      x        2            −                        x          4                2            +                        x          6                3            +      ⋯      Dividing both numerator and denominator by       x    2   gives the result  L  =  −      7    3

Find the limit given that f ( 1 ) = 1 , f ( x + y ) = f ( x )

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