Prove $\lim_{n\to\infty}\frac{2n}{n+2}=2$

Question

Prove $$\lim_{n\to\infty} \frac{2n}{n+2} = 2$$

I know for all $\epsilon > 0$ there exists some $k\in\mathbb{N}$ for all $n > K$ and $\left|\frac{2n}{n+2} - 0\right| < \epsilon$.

Proof:

Fix any $\epsilon > 0$ choose $k \in \mathbb{N}$ where there exists $???? < K$

I am getting stuck on what to choose for $K$ in terms of $\epsilon$

I know that $\frac{2n}{n+2} < \frac{2n}{n} = 2$ but from there I get stuck.

Answer 1

Let $\varepsilon>0$. Write $\displaystyle \frac{2n}{n+2}$ as $\displaystyle 2-\frac{4}{n+2}$. You have:

$$\left|\left(2-\frac{4}{n+2}\right)-2\right|=\frac{4}{n+2}$$

So for $n>\frac{4}{\varepsilon}-2$ you have $\frac{4}{n+2}<\varepsilon$.

Answer 2

$$\lim_{n\to\infty }\frac{2n}{n+1}=\lim_{n\to\infty }\frac{2n}{n\left(1+\frac{1}{n}\right)}=2$$

Answer 3

We want to show that for any $\epsilon\gt 0$ there is an integer $K=K(\epsilon)$ such that if $n\gt K$ then $$\left|\frac{2n}{n+2}-2\right|\lt \epsilon.\tag{1}.$$ So let $\epsilon\gt 0$ be given. The left-hand side of (1) simplifies to $$\frac{4}{n+2}.$$ So to satisfy the inequality (1), it is enough to let $n$ be a positive integer such that $\frac{4}{n+2}\lt \epsilon$, or equivalently $n+2\gt \frac{4}{\epsilon}$.

In particular, let $K=\left\lfloor \frac{4}{\epsilon}\right\rfloor$. If $n\gt K$, then $n+2\gt \frac{4}{\epsilon}$, and therefore Inequality (1) holds.

Answer 4

$\textbf{Proof:}$

Let $\epsilon >0$ be given.

$\textbf{RTP:}$ $\exists N_\epsilon \in \mathbb{N} \ni |\frac{2n}{n+2} -2|<\epsilon \ \forall \ n \geq N_\epsilon $.

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$\textbf{ROUGH WORK:}$

We know that $|\frac{2n}{n+2}-2| = |\frac{-4}{n+2}| =\frac{4}{n+2} \forall n\in \mathbb{N}$.

Now we need $\frac{4}{n+2} < \epsilon \implies \frac{n+2}{4} > \frac{1}{\epsilon}$ (Since all values are positive) $\implies n+2 > \frac{4}{\epsilon} \implies n> \frac{4}{\epsilon}-2$.

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For the given $\epsilon > 0$, by the Archimedean Property, we may choose $N_\epsilon > \frac{4}{\epsilon}-2$.

We now have, $\forall n \geq N_\epsilon$ :

\begin{align*} n \geq N_\epsilon > \frac{4}{\epsilon} -2\end{align*} $ \implies n+2 \geq N_\epsilon +2 > \frac{4}{\epsilon} \\ \implies \frac{n+2}{4} \geq \frac{N_\epsilon +2}{4} > \frac{1}{\epsilon} \\ \implies \frac{4}{n+2} \leq \frac{4}{N_\epsilon +2}<\epsilon \\ \implies \frac{4}{n+2}<\epsilon$

And hence, $\lim_{n\to \infty}(\frac{2n}{n+2})=2$