# How to use ε-N language to prove $\lim_{n\to ∞}\frac{\sqrt[3]{n^2}\sin n}{n+1}=0$ [closed]

The statement

$$\lim_{n\to ∞}\frac{\sqrt[3]{n^2}\sin n}{n+1}=0$$ is equivalent to

$$\forall \epsilon > 0 \quad \exists N_\epsilon \quad \mathrm{s.t.} \quad \forall n \geq N_\epsilon \quad \left|\frac{\sqrt[3]{n^2}\sin n}{n+1}-0 \right|\leq \epsilon.$$

I don't know how to express $N_\epsilon$ in terms of $\epsilon$ in this question.

## closed as unclear what you're asking by qbert, user99914, Namaste, Ethan Bolker, José Carlos SantosJul 7 '18 at 20:35

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• There's no $x$ in your formula.... Do you mean $\lim_{n\to\infty}\cdots$? – Lord Shark the Unknown Jul 7 '18 at 2:11
• @LordSharktheUnknown sorry, I made the mistake when I paste the code.... – Chloe Zhou Jul 7 '18 at 2:14
• Can you use squeeze theorem? That would be a lot easier. – user122049 Jul 7 '18 at 2:19

Notice that, for all $n$, because $|\sin n| \leq 1$ and $1/|n+1| \leq 1/|n|$, $$|f(n)| =\left|\frac{n^{2/3} \sin n}{n+1}\right| \leq \left|\frac{n^{2/3}}{n+1}\right| \leq \left| \frac{n^{2/3}}{n}\right| = \left| \frac 1 {n^{1/3}} \right|.$$ So $|f(n)| \leq \epsilon$ when $\left|n^{-1/3} \right|\leq \epsilon$, that is, when $$n \geq \frac 1 {\epsilon^3} =: N_\epsilon.$$

• Would the down-voter care to explain himself/herself? I did nothing short of answering the OP's question. – giobrach Jul 7 '18 at 2:56

Try to decompose the formula and see how the components behave individually as $n \to \infty$. How does $\sqrt[3]{n^2}$ behave? What about $\frac{1}{n+1}$? What effect do the oscillations of $\sin n$ have on the terms of the sequence (what is the limited range of $\sin n$)?

• But in that case, am I still using the N-e language? I thought the question require me to express n in terms of e – Chloe Zhou Jul 7 '18 at 2:16
• @ChloeZhou Yes! But before you dive into finding the $n$ for your $\varepsilon$, try to think how the sequence behaves in terms of the components. Why do the terms get arbitrarily close to $0$? – Andrey Portnoy Jul 7 '18 at 2:19
• I think you didn't get the question – Gaston Burrull Jul 7 '18 at 2:25
• @GastónBurrull I'm referring you to Theorem 3.3c in Rudin's PMA. My idea is to give hints, as this is clearly a homework problem, and not just give away the solution. – Andrey Portnoy Jul 7 '18 at 2:29