Science:Math Exam Resources/Courses/MATH101 C/April 2024/Question 09/Hint 1

For a more elementary solution that doesn't use the limit comparison test, consider the following rewriting of the integrand:

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

Can you use this to show that the integrand is bounded by ${\frac {K}{x}}$ for some $K$?

Alternatively, the second solution applies the limit comparison test.