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

Salvador Bush 2022-07-12 Answered
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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Answers (1)

conveneau71
Answered 2022-07-13 Author has 17 answers
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
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