How to write this delta-epsilon proof?

Question

I was wondering if anyone could show me how to do this proof? I can't seem to get it into a simple form.

Prove that $\lim\limits_{x \to 2}\frac{1}{x} = \frac{1}{2}$.

Answer 1

Let's first require that $|x-2|\le1$ so that $1\le x\le3$.

This implies that $\frac{1}{|x|}\le1$.

Therefore,

$$\begin{align} \left|\frac1x-\frac12\right|&=\frac12 \frac{|x-2|}{|x|}\\\\ &\le \frac12 |x-2|\\\\ &<\epsilon \end{align}$$

whenever $|x-2|<\delta =\min\left(1,2\epsilon\right)$

Answer 2

Let $\epsilon>0$ be arbitrary.

Now let $\displaystyle \delta =$min$\{1,2\epsilon\}$

Then $|x−2|≤1 \Rightarrow 1≤x≤3 \Rightarrow \frac{1}{|x|}\le 1$.

Observe that $\displaystyle \left|\frac{1}{x}-\frac{1}{2}\right|=\left|\frac{x-2}{2x}\right|=\frac{|x-2|}{2|x|}\le \frac{\delta}{2}\le \epsilon$