A limit to infinity with four different roots: $\lim_{n \to \infty} \frac{\sqrt{n}+ \sqrt[4]{n^2+n}}{\sqrt[6]{n^3-1}+\sqrt[3]{n+2}}$

Question

$$\lim_{n \to \infty} \frac{\sqrt{n}+ \sqrt[4]{n^2+n}}{\sqrt[6]{n^3-1}+\sqrt[3]{n+2}}$$

Okay, this is the last limit I have to solve but it's not easy at all. It has four different roots and I have no idea what to do. I've tried everything :(

I would be really grateful if someone could solve this! :) Thank you in advance! :)

Answer 1

Hint: $$\dfrac{\sqrt{n}+\sqrt[4]{n^2+n}}{\sqrt[6]{n^3-1}+\sqrt[3]{n+2}} = \dfrac{1+\sqrt[4]{1+\dfrac1{n}}}{\sqrt[6]{1-\dfrac1{n^3}}+\sqrt[6]{\dfrac1{n}+\dfrac4{n^2}+\dfrac4{n^3}}}.$$