I assume the denominator is [tex]x-2[/tex].
[tex]\lim_{x\to2}\dfrac{\dfrac{1}{x}-\dfrac{1}{2}}{x-2}=\\
\lim_{x\to2}\dfrac{\dfrac{2}{2x}-\dfrac{x}{2x}}{x-2}=\\
\lim_{x\to2}\dfrac{\dfrac{2-x}{2x}}{x-2}=\\
\lim_{x\to2}\dfrac{2-x}{2x(x-2)}}=\\
\lim_{x\to2}-\dfrac{x-2}{2x(x-2)}}=\\
\lim_{x\to2}-\dfrac{1}{2x}}=\\
-\dfrac{1}{2\cdot2}=\\
-\dfrac{1}{4}=
[/tex]
