Example 11. Find $\lim _{n \rightarrow \infty} \frac{1+3+5+\ldots+(2 n-1)}{1+2+3+\ldots+n}$.
Solution. The numerator and denominator of the fraction are the sums of $n$ terms of the corresponding arithmetic progressions. Finding these sums using the known formula, we get

\[
\begin{gathered}
\lim _{n \rightarrow \infty} \frac{1+3+5+\ldots+(2 n-1)}{1+2+3+\ldots+n}=\lim _{n \rightarrow \infty} \frac{\frac{1+(2 n-1)}{2} \cdot n}{\frac{1+n}{2} \cdot n}= \\
=\lim _{n \rightarrow \infty} \frac{1+(2 n-1)}{1+n}=\lim _{n \rightarrow \infty} \frac{2 n}{n+1}=2 \lim _{n \rightarrow \infty} \frac{n}{n+1}=2 \lim _{n \rightarrow \infty} \frac{1}{1+\frac{1}{n}}=2 .
\end{gathered}
\]