Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 09/Hint 2

Note first that, since the denominator does not vanish on the interval $[1,\infty )$, the integral is improper because one of the integration bounds is infinity (as opposed to there being a singular point in the domain of integration).

Now the integrand can be approximated as follows for large $x$:

$${\frac {x^{2}+1}{x^{3}+2}}\approx {\frac {x^{2}}{x^{3}}}={\frac {1}{x}}.$$

Can you use this approximation together with the Limit Comparison test?