# How would you find the roots to this question?

I have a homework problem that I arrived.

With Mathematica, the limit is 0. So by using $\epsilon= 10^{-6}$ (it is -6, not -0, sorry for the cutoff).

$\sin(n^2)/\sqrt{n} <\epsilon =10^{-6}$

So I tried putting that ino Mathematica and no luck so I have a feeling I am approaching this problem the wrong way

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Hint: $\frac{sin(n^{2})}{\sqrt{n}}\leq\frac{1}{\sqrt{n}}$ and the latter is monotonic so it suffices to find an $N$ s.t $\frac{1}{\sqrt{N}}<\epsilon$
How do you know if the sequence $1/\sqrt{n}< \epsilon$ in the first place (the same epsilon)? – Hawk Sep 22 '12 at 23:59
Find such $N$. Hint: find $N$ s.t $\sqrt{N}>\frac{1}{\epsilon}$ (why ?) – Belgi Sep 23 '12 at 0:01
You should get $10^{12} = N$ – Hawk Sep 23 '12 at 0:03
We have $|\sin \theta|\le 1$ for all $\theta$, so if we can make $\frac{1}{\sqrt{n}}\lt \epsilon$, then for sure $\left|\frac{\sin(n^2)}{\sqrt{n}}\right|\lt \epsilon$. – André Nicolas Sep 23 '12 at 1:39