<s>[INST] When does $\int_a^\infty g(x)/f(x)\,dx$ converge? 
When does $\int_a^\infty g(x)/f(x) \,dx$ converge?

Where $g$ and $f$ are polynomials.
I am not so sure about this pattern. Consider,
$\int_1^\infty \frac{1}{x^3 + x^2 + x + 1} \, dx$, this integral converges, but $\int_1^\infty \frac{4x^2}{x^3 + x^2 + x + 1} \,dx$ does not converge.
How can I tell when it converges? And what is the pattern?
 [/INST] In the integral $\displaystyle \int_1^\infty \frac{4x^2}{x^3 + x^2 + x + 1} \,dx$ the leading terms in the numerator and denominator are $4x^2$ and $x^3$, so as $x\to\infty$ it goes to $0$ at a rate comparable to that  of $4x^2/x^3 = 4/x$, so the integral diverges.
$$
\frac{4x^2}{x^3+x^2+x+1} = \frac 4 {x + 1 + \dfrac 1 x + \dfrac 1 {x^2}} \ge  \frac 1 x \text{ if } x\ge 1, \text{ and } \int_1^\infty \frac{dx} x = \infty.
$$
By contrast $\dfrac 1 {x^3+x^2+x+1} \le \dfrac 1 {x^2}$ and $\displaystyle\int_1^\infty \dfrac{dx}{x^2}<\infty.$
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