Příklad 2.1

Spočtěte limitu posloupnosti \(\large\lim_{n\to\infty}\frac{1+2n+5n^2}{2n-n^2}\)

$$\Large \begin{aligned}

& \lim_{n\to\infty}\frac{1+2n+5n^2}{2n-n^2} \

=& \lim_{n\to\infty}\frac{n^2\cdot(\frac{1}{n^2}+\frac{2}{n}+5)}{n^2\cdot(\frac{2}{n}-1)} \

=& \lim_{n\to\infty}\frac{\cancel{n^2}^1\cdot(\frac{1}{n^2}+\frac{2}{n}+5)}{\cancel{n^2}^1\cdot(\frac{2}{n}-1)} \

=& \lim_{n\to\infty}\frac{\frac{1}{n^2}+\frac{2}{n}+5}{\frac{2}{n}-1} \

=& \frac{\lim_{n\to\infty}(\frac{1}{n^2}+\frac{2}{n}+5)}{\lim_{n\to\infty}(\frac{2}{n}-1)} \

=& \frac{\lim_{n\to\infty}(\frac{1}{n^2})+\lim_{n\to\infty}(\frac{2}{n})+\lim_{n\to\infty}(5)}{\lim_{n\to\infty}(\frac{2}{n})-\lim_{n\to\infty}(1)} \

=& \frac{0+0+5}{0-1} \

=& \frac{+5}{-1} \

=& \boxed{-5}

\end{aligned} $$