Rational functions and residues

Question

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 1

Consider $$f(x)=\frac{x^{100}+1}x$$ and $$g(x)=\frac{x^2+1}x$$

They both have simple pole at $0$.

The residues are both $1$.

They both tends to $\infty$ as $x\to \infty$.

But $$f(x)\ne g(x)$$

p.s. I am not sure what you mean by ‘same behaviour’. Do you specifically mean $$\lim_{x\to\infty}\frac{f(x)}{g(x)}=1$$?

Even you do mean that, just consider $$\frac{x^2+x+1}x$$ and $$\frac{x^2+1}x$$