Answer
$\dfrac{1}{x+2}$
Work Step by Step
The given expression, $
\dfrac{\dfrac{x-1}{x^2-4}}{1+\dfrac{1}{x-2}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{\dfrac{x-1}{x^2-4}}{\dfrac{x-2+1}{x-2}}
\\\\=
\dfrac{\dfrac{x-1}{x^2-4}}{\dfrac{x-1}{x-2}}
\\\\=
\dfrac{x-1}{x^2-4}\div\dfrac{x-1}{x-2}
\\\\=
\dfrac{x-1}{x^2-4}\cdot\dfrac{x-2}{x-1}
\\\\=
\dfrac{x-1}{(x+2)(x-2)}\cdot\dfrac{x-2}{x-1}
\\\\=
\dfrac{\cancel{x-1}}{(x+2)(\cancel{x-2})}\cdot\dfrac{\cancel{x-2}}{\cancel{x-1}}
\\\\=
\dfrac{1}{x+2}
.\end{array}
