Prove that $\lim \limits_{n\to\infty}\frac{n}{n^2+1}  = 0$ from the definition This is a homework question:

Prove, using the definition of a limit, that
  $$\lim_{n\to\infty}\frac{n}{n^2+1}  = 0.$$

Now this is what I have so far but I'm not sure if it is correct:
Let $\epsilon$ be any number, so we need to find an $M$ such that:
$$\left|\frac{n}{n^2 + 1}\right| < \epsilon \text{ whenever }x \gt M.$$
$$ n \lt \epsilon(n^2 + 1) $$
$$n \lt  \epsilon n^2 + \epsilon$$
Now what? 
I am completely clueless on how to do this!
 A: You had the right start.  We want to show that for any $\epsilon>0$, there exists an integer $N$ such that $\left|\frac{n}{n^2+1}-0\right| <\epsilon$ whenever $n>N$.
Or alternately, we want to show that there is an $M$, not necessarily an integer, such that $\left|\frac{n}{n^2+1}-0\right| <\epsilon$ whenever $n>M$.
We solve the problem by producing a suitable $N$ (alternately, $M$).
So let $\epsilon>0$ be given.  Note that if $n>0$ then
$$\left|\frac{n}{n^2+1}-0\right|=\frac{n}{n^2+1}<\frac{n}{n^2}=\frac{1}{n}.$$
So if we choose $n$ positive, it is enough to make sure that $\frac{1}{n}<\epsilon$. This inequality is easy to solve for $n$.
The inequality can be rewritten as $n >\frac{1}{\epsilon}$. It will hold if $n > \lceil \frac{1}{\epsilon}\rceil$.  So we can choose, for example, $N=\lceil\frac{1}{\epsilon}\rceil$.  Here by $\lceil x \rceil$ we mean the smallest integer which is $\ge x$.
Alternately and more simply, if $M=\frac{1}{\epsilon}$, then the desired inequality holds whenever $n>M$.
We have shown that given $\epsilon>0$, if we choose  $N=\lceil\frac{1}{\epsilon}\rceil$, then $\left|\frac{n}{n^2+1}-0\right|<\epsilon$ whenever $n>N$. By definition we therefore have
$$\lim_{n\to \infty}\frac{n}{n^2+1} =0.$$
Comment: There is no need to solve the inequality $\frac{n}{n^2+1}<\epsilon\:$ exactly. We are not being asked to find the cheapest $N$ such that if $n >N$, then the inequality holds. All we are asked to do is to show that there is such an $N$.  It can be very helpful, in this case and elsewhere, to replace our expression $\frac{n}{n^2+1}$ by something which is guaranteed to be larger (but still approaches $0$), and is substantially simpler. In the solution we replaced  $\frac{n}{n^2+1}$  by $\frac{1}{n}$. This saved us from the slightly painful task of solving a quadratic inequality.
A: First, $\epsilon$ should not be "any number", it should be "any positive number."
Now, you are on the right track. What do you need in order for $\frac{n}{n^2+1}$ to be smaller than $\epsilon$? You need $n\lt \epsilon n^2 + \epsilon$. This is equivalent to requiring
$$\epsilon n^2 - n + \epsilon \gt 0.$$
You want to find out for what values of $n$ this is true. This is a quadratic inequality: you first solve
$$\epsilon n^2 - n + \epsilon = 0,$$
and then you use the solution to determine where the quadratic is positive, and where it is negative. The answer will, of course, depend on $\epsilon$.
Using the quadratic formula, we have that
$$\epsilon n^2 - n + \epsilon = 0$$
has solutions
$$n = \frac{1 + \sqrt{1-4\epsilon^2}}{2\epsilon}, \quad n= \frac{1-\sqrt{1-4\epsilon^2}}{2\epsilon}.$$
That is,
$$\epsilon n^2 - n + \epsilon = \epsilon\left( n - \frac{1-\sqrt{1-4\epsilon^2}}{2\epsilon}\right)\left(n - \frac{1+\sqrt{4\epsilon^2}}{2\epsilon}\right).$$
Now, we can assume that $\epsilon\lt \frac{1}{2}$, so that $4\epsilon^2\lt 1$ (if it works for all small enough $\epsilon$, then it works for all $\epsilon$. Since $\epsilon\gt 0$, then the quadratic is positive if $n$ is smaller than the smallest of the roots, or if $n$ is larger than the larger of the two roots. The larger root is $\displaystyle \frac{1 + \sqrt{1-4\epsilon^2}}{2\epsilon}$. So if
$$n \gt \frac{1+\sqrt{1-4\epsilon^2}}{2\epsilon},$$
then
$$\epsilon n^2 -n + \epsilon \gt 0.$$
Can you finish it up from here?
A: You could harvest from
$\frac{n}{n^2+1} = \frac{1}{n+\frac{1}{n}}$ for any $n$ in $\mathbb{R}^\star$, and for any $\varepsilon>0,n>0$ then $\frac{1}{n+\frac{1}{n}}<\varepsilon \Leftrightarrow n + \frac{1}{n} > \frac{1}{\varepsilon}$.
On the range $\left[1;+\infty\right[$ then $\frac{1}{n}$ is upper-bounded by $1$.
A: $n<\epsilon n^2 +\epsilon$ if and only if $n^2 - \frac{n}{\epsilon} + 1 > 0$, if and only if $(n - \frac{1}{2\epsilon})^2 + (1 - \frac{1}{4 \epsilon^2}) > 0$ (complete the square)
You should be able to do it from here!  Good luck.
