5.2. Find the limit of the sequence

$$
\lim _{n \rightarrow \infty} \frac{\sqrt{2 n-1}-\sqrt[3]{n^{3}+10}}{\sqrt{n^{2}+1}-\sqrt[4]{n+1}}
$$
Solution. Divide the numerator and the denominator of the fraction by $n$. After transformations and discarding infinitely small quantities, we obtain the required result:

$$
\begin{gathered}
\lim _{n \rightarrow \infty} \frac{\sqrt{2 n-1}-\sqrt[3]{n^{3}+10}}{\sqrt{n^{2}+1}-\sqrt[4]{n+1}}=\lim _{n \rightarrow \infty} \frac{\sqrt{2 n-1} / n-\sqrt[3]{n^{3}+10} / n}{\sqrt{n^{2}+1} / n-\sqrt[4]{n+1} / n}= \\
=\lim _{n \rightarrow \infty} \frac{\sqrt{2 / n-1 / n^{2}}-\sqrt[3]{1+10 / n^{3}}}{\sqrt{1+1 / n^{2}}-\sqrt[4]{1 / n^{3}+1 / n^{4}}}=-1
\end{gathered}
$$