Suppose you have two rational functions of x having the

Theresa Archer

Theresa Archer

Answered question

2022-06-24

Suppose you have two rational functions of x having the same (finite) limit when x , having the same simple poles and the same residues in these poles, does it imply that they're equal? How to show that ?

Answer & Explanation

Zayden Andrade

Zayden Andrade

Beginner2022-06-25Added 22 answers

Consider
f ( x ) = x 100 + 1 x
and
g ( x ) = x 2 + 1 x
They both have simple pole at 0.
The residues are both 1.
They both tends to as x .
But
f ( x ) g ( x )
p.s. I am not sure what you mean by ‘same behaviour’. Do you specifically mean
lim x f ( x ) g ( x ) = 1?
Even you do mean that, just consider
x 2 + x + 1 x
and
x 2 + 1 x

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