Let f ( x , y ) = ( x 2 </msup> + y 2 </msup> + 2

jistefaftexia99kq6 2022-05-14 Answered
Let f ( x , y ) = ( x 2 + y 2 + 2 ) x y defined on R 2 and let g ( x , y ) be a degree 2 polynomial in two variables.
Suppose lim ( x , y ) ( 0 , 0 ) g ( x , y ) f ( x , y ) x 2 + y 2 = 0 . Then, determine g ( x , y )
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partyjnopp9wa
Answered 2022-05-15 Author has 17 answers
Recall that for e t = 1 + t + O ( t 2 ) and ln ( 1 + t ) = t + o ( t ). Therefore,for ( x , y ) ( 0 , 0 )
( x 2 + y 2 + 2 ) x y = exp ( x y ln ( x 2 + y 2 + 2 ) ) = 1 + x y ln ( x 2 + y 2 + 2 ) + O ( x 2 y 2 ) = 1 + x y ( ln ( 2 ) + ln ( 1 + x 2 + y 2 2 ) ) + o ( x 2 + y 2 ) = 1 + x y ( ln ( 2 ) + x 2 + y 2 2 + o ( x 2 + y 2 ) ) + o ( x 2 + y 2 ) = 1 + ln ( 2 ) x y + o ( x 2 + y 2 ) .
Hence g ( x , y ) = 1 + ln ( 2 ) x y. In particular a 3 = ln ( 2 )
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