When there are some conditions on the power of a rational function, how can we evaluate/prove the no

Gybrisysmemiau7

Gybrisysmemiau7

Answered question

2022-06-21

When there are some conditions on the power of a rational function, how can we evaluate/prove the non-existence of the limit of the function? Here is a specific example:
Prove
lim ( x , y ) ( 0 , 0 ) ( x y ) p 1 ( ( p 2 ) x 2 + 2 x y + p y 2 ) ( x 2 + y 2 ) 2 = 0
if p > 3
Is this even solvable?

Answer & Explanation

Savanah Hernandez

Savanah Hernandez

Beginner2022-06-22Added 16 answers

The expression is bounded above by
( | x | + | y | ) p 1 ( ( p 2 ) | x | 2 + 2 | x | | y | + p y 2 ) ( x 2 + y 2 ) 2 .
Let r = ( x 2 + y 2 ) 1 / 2 . Use | x | , | y | r to finish.

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