<s>[INST] Please help with this limit of a sequence: $\lim\limits_{n\to\infty} \bigg(n\sqrt[3]{1+\frac1n} - n\sqrt[3]{1+\frac1{n^2}}\bigg)$ I have this limit of a sequence: $$\lim_{n\to\infty} \bigg(n\sqrt[3]{1+\frac1n} - n\sqrt[3]{1+\frac1{n^2}}\bigg).$$
Can I modify this expression to this expression by knowing that fractions goes to $0$?: $\lim n-n$. How to modify this solution to get the limit?
 [/INST] Let $a=\sqrt[3]{n^3+n^2},\,b=\sqrt[3]{n^3+n}$ so $a-b=\frac{a^3-b^3}{a^2+ab+b^2}=\frac{n^2-n}{a^2+ab+b^2}\approx\frac{n^2}{n^2+n^2+n^2}=\frac13$.
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