# \$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?