$lim_{x to 1} frac{1}{x+2} = 1/3$ with $varepsilon$-$delta$ definition?

I’m trying to use the $varepsilon$$delta$ definition of a limit to prove that
$$
lim_{x to 1} frac{1}{x+2} = 1/3.
$$

But I’m getting stuck on finding the correct $delta$. Here is my try:

begin{align*}
lvert f(x) – L rvert < varepsilon \
lvert frac{1}{x+2} – frac{1}{3} rvert < varepsilon \
lvert frac{x -1}{2 – x} rvert < 3varepsilon.
end{align*}

And then I’m not really sure what to do. How do you proceed from here?