Check whether $$\int_{-\infty}^{\infty}\frac1{\sqrt{x^{10}+2}}\ dx$$ converges or diverges.
I used the Limit Comparison Test with the function $\frac1{x^5}$: $$\lim_{x\to\infty}\frac{\sqrt{x^{10}+2}}{x^5}=1$$ Also, $$\int_{-\infty}^{\infty}\frac1{x^5}\ dx=\infty$$ Therefore, it means that:$$\int_{-\infty}^{\infty}\frac1{\sqrt{x^{10}+2}}\ dx=\infty$$ However, the latter integral converges. What is wrong with this approach?