Let's say we have this limit: $$\lim\limits_{x\to \infty} \frac{1}{x}$$ which is clearly $$\lim\limits_{x\to \infty} \frac{1}{x} = 0.$$ From there, to prove it we should: $$\left\lvert \frac{1}{x} - 0 \right\rvert < \epsilon$$ (with $\epsilon > 0$ and small).
To solve that inequality we should deal with a system of: $$\begin{align*} \frac{1}{x} &\lt \epsilon&&\text{(for }\frac{1}{x} \gt 0\text{)}\\ \frac{1}{x} &\gt -\epsilon&&\text{(for }\frac{1}{x}\lt 0\text{)} \end{align*}$$ Then from the ()
we have that the first inequality is for $x < 0$ and the second is for $x > 0$.
Is this right?