Potentiometer min and max value
Well, you kind of obviously can:







simulate this circuit &ndash; Schematic created using CircuitLab

If your potentiometer allows values in \$[a,b]\$, then your maximum resistance is

\begin{align}
R_{\text{max}} &amp;= \frac1{\frac1{R_{\text{series}}+b}+\frac1{R_{\text{parallel}}}}
\end{align}

and likewise, your minimum resistance is

\begin{align}
R_\text{min} &amp;= \frac1{\frac1{R_\text{series}+a}+\frac1{R_\text{parallel}}}
\end{align}

Solving for the different resistors is left as an exercise to the reader.

However, since the minimum value probably should be orders smaller than the maximum value, things simplify to

\begin{align}
R_\text{min} &amp;\approx R_\text{series}\\
R_\text{max} &amp;\approx R_\text{parallel}||b
\end{align}