<s>[INST] Does $\lim_{n\to \infty} \left(\frac{a}{a+bn}\right)^{\frac{1}{n}}$ exist? Let $a, b >0$ and $a \ll b$, $a<1$. Does $\lim_{n\to \infty} \left(\frac{a}{a+bn}\right)^{\frac{1}{n}}$ exist ?
 [/INST] Using
\begin{align}
\frac{a}{a + b \, n} &= 1 - \frac{1}{1 + \frac{a}{b \, n}} \\
&= 1 - \left( 1 - \frac{a}{b \, n} + \frac{a^{2}}{b^{2} \, n^{2}} + \mathcal{O}\left(\frac{1}{n^{3}}\right) \right) \\
&= \frac{a}{b \, n} - \left(\frac{a}{b \, n}\right)^{2} + \mathcal{O}\left(\left(\frac{a}{b \, n}\right)^{3}\right) \\
&= \frac{a}{b \, n} \, \left( 1 - \frac{a}{b \, n} + \mathcal{O}\left(\left(\frac{a}{b \, n}\right)^{2}\right) \right)
\end{align}
then
\begin{align}
\lim_{n \to \infty} \left( \frac{a}{a + b \, n} \right)^{\frac{1}{n}} &= \lim_{n \to \infty} \, \left(\frac{a}{b \, n}\right)^{\frac{1}{n}} \, \left( 1 - \frac{a}{b \, n} + \mathcal{O}\left(\left(\frac{a}{b \, n}\right)^{2}\right) \right)^{\frac{1}{n}}.
\end{align}
By inverting $n$, ie $n \to \frac{1}{n}$, the limit becomes
\begin{align}
\lim_{n \to \infty} \left( \frac{a}{a + b \, n} \right)^{\frac{1}{n}} &= \lim_{n \to 0} \, \left(\frac{a \, n}{b}\right)^{n} \, \left( 1 - \frac{a \, n}{b} + \mathcal{O}\left(\left(\frac{a \, n}{b}\right)^{2}\right) \right)^{n} = 0^{0} = 1.
\end{align}
</s>