True or False. The domain of every rational function is the set of all real numbers.

True or False. The graph of a rational function may intersect a horizontal asymptote.

Does there exists some simple criteria to know when the primitive of a rational function of ${\displaystyle \mathbb{C}\left[z\right]}$ is still a rational function?
In fact my question is more about the stability of this property. Let P and Q two co' polynomials and let A and B two co' polynomials such that
$\frac{A}{B}=(\frac{P}{Q}{)}^{\prime }=\frac{P^{\prime }Q-P{Q}^{\prime }}{Q^2}$
Then considering a pertubation ${\displaystyle A^{\xi }}$ of A (in the sense the roots of ${\displaystyle A^{\xi }}$ converge to the one of A with the same multiplicity as ${\displaystyle \xi }$ goes to zero.): does ${\displaystyle \frac{A^{\xi }}{B}}$ admit a primitive which is rational function?

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits.
${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}\frac{\sqrt{x^2+1}}{x+1}}$

Rational number: wha is 0.02 as a rational number

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits. $\underset{x\to \mathrm{\infty }}{lim}\frac{((x^{-1})+(x^{-4}))}{((x^{-2})-(x^{-3}))}$

If I have two algebraic numbers $\alpha$ and $\beta$ and a rational function $w$ with rational coefficients (a function that's the ratio of two rational polynomials) that relates the two $\alpha =w(\beta )$ if I were to substitute $\beta$ with one of it's algebraic cojugates ${\beta }^{{}^{\prime }}$ in the rational function will I get a conjugate of $\alpha$ or rather is $w({\beta }^{{}^{\prime }})$ an algebraic conjugate of alpha? I feel like there is a very obvious counter example but I've been struggling to find one.

I have
${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{(-1)}^{n}2n{x}^{(2n-1)}}$
It turns out that this series is equal to the function ${\displaystyle \frac{-2x}{{(1+x^2)}^2}}$
Is there a general method that would demonstrate this fact beforehand?