Definition of limits

Question

I have this following equation that I and my classmates have different opinions on. We're supposed to prove the limits using the formal definition.

Here is the question, and my method of approaching it. Please tell me if I am doing it wrong or not approaching it the right way. P.S: If you are about to tell me to ask my professor, well... He doesn't like to help that much.

Answer 1

You may suppose $x>0$. Then the condition to be satisfied is $$\biggl|\frac 1x-\frac12\biggr|=\biggl|\frac{x-2}{2x}\biggr|<\varepsilon\iff|x-2|<2\,\varepsilon |x|<2\,\varepsilon(|x-2|+2)=2\,\varepsilon|x-2|+4\varepsilon, $$ so , if $\;\varepsilon<\frac12 $, we obtain $$|x-2|<\frac{4\varepsilon}{1-2\varepsilon}.$$

Answer 2

if you factor a $1/2$ out of a $-\frac{1}{2}$, you are left with $-1$, not $4$.

If you factor a $1/2$ out of a term $\frac{1}{x}$, you are left with $\frac{1}{2x}$, not $2x$.