characteristic impedance

at every section of $\infty$ length transmission line, looking forward/backward, we see an impedance that is called characteristic impedance of a line .... for a lossless line (inductive and capacitive only):

$$\begin{eqnarray} Z_0 &=& jlw + \frac{1}{\frac{1}{Z_0}+jcw} \nonumber \\ jlw+\frac{Z_0}{1+jcwZ_0} &=& Z_0 \nonumber \\ j \left(lw-Z_0^2 cw\right) - lcw^2 z_0 &=& 0 \label{eq:1} \\ \end{eqnarray}$$ for a very short piece of T-line (length approaches zero), we can assume $l \times c \rightarrow 0$. In this case, equation $\eqref{eq:1}$ results in: $$\begin{eqnarray} Z_0 &=& \sqrt{\frac{l}{c}} \end{eqnarray}$$ $Z_0$ is length independent and is called characteristic impedance of a transmission line.