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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:

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- Evaluate the limit $\lim\limits_{n \to \infty} \frac{1}{1+n^2} +\frac{2}{2+n^2}+ \ldots +\frac{n}{n+n^2}$

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!

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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.

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?

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 $1$.

$n<\epsilon n^2 +\epsilon$ 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.

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