<s>[INST] Evaluate $\lim_{n\to \infty} (n+1)^{\frac 23} -(n-1)^{\frac 23}$ Clearly, $\infty -\infty \not =0$
I have a feeling the squeeze theorem can be applied here, but I am not sure how to write the required terms separately. Can I get a hint?
 [/INST] Let $f:[n-1,n+1]\to\mathbb{R}$ and $f(x)=x^{\frac{2}{3}}$. Then from Mean value theorem (Lagrange’a) exist $\xi\in (n-1,n+1)$ such that:
$$\frac{f(n+1)-f(n-1)}{n+1-(n-1)}=f'(\xi)$$
but (check it out)
$$\frac{2}{3 \sqrt[3]{n+1} }\le f'(\xi)\le \frac{2}{3 \sqrt[3]{n-1} } $$
so
$$\frac{4}{3 \sqrt[3]{n+1} }\le f(n+1)-f(n-1 )\le \frac{4}{3 \sqrt[3]{n-1} } $$
$$\frac{4}{3 \sqrt[3]{n+1} }\le (n+1)^{\frac{2}{3}}-(n-1 )^{\frac{2}{3}}\le \frac{4}{3 \sqrt[3]{n-1} } $$
now You can aplay Squeeze theorem.
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