1. C.

Since $a_{n}=\frac{S_{2 n-1}}{2 n-1}, b_{n}=\frac{T_{2 n-1}}{2 n-1}$, then $\frac{a_{n}}{b_{n}}=\frac{S_{2 n-1}}{T_{2 n-1}}=\frac{4 n-2}{6 n-2}=\frac{2 n-1}{3 n-1}$. Therefore, $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\frac{2}{3}$.
1. For the arithmetic sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$, the sums of the first $n$ terms are $S_{n}$ and $T_{n}$, respectively. If $\frac{S_{n}}{T_{n}}=\frac{2 n}{3 n+1}$, then $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}$ equals ( ).
A. 1
B. $\frac{\sqrt{6}}{3}$
C. $\frac{2}{3}$
D. $\frac{4}{9}$