zbadaj granice ciągu
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Witam!
Zauważ, że , stąd
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Witam!
Zauważ, że
, stąd ![\lim_{n\to\infty}{\frac{n(n+1)}{2(n^2+2n+5)}}=\lim_{n\to\infty}{\frac{n^2+n}{2n^2+4n+10}}=\lim_{n\to\infty}{\frac{1+\frac{1}{n}}{2+\frac{4}{n}+\frac{10}{n^2}}}=\frac{1}{2} \lim_{n\to\infty}{\frac{n(n+1)}{2(n^2+2n+5)}}=\lim_{n\to\infty}{\frac{n^2+n}{2n^2+4n+10}}=\lim_{n\to\infty}{\frac{1+\frac{1}{n}}{2+\frac{4}{n}+\frac{10}{n^2}}}=\frac{1}{2}](https://tex.z-dn.net/?f=%5Clim_%7Bn%5Cto%5Cinfty%7D%7B%5Cfrac%7Bn%28n%2B1%29%7D%7B2%28n%5E2%2B2n%2B5%29%7D%7D%3D%5Clim_%7Bn%5Cto%5Cinfty%7D%7B%5Cfrac%7Bn%5E2%2Bn%7D%7B2n%5E2%2B4n%2B10%7D%7D%3D%5Clim_%7Bn%5Cto%5Cinfty%7D%7B%5Cfrac%7B1%2B%5Cfrac%7B1%7D%7Bn%7D%7D%7B2%2B%5Cfrac%7B4%7D%7Bn%7D%2B%5Cfrac%7B10%7D%7Bn%5E2%7D%7D%7D%3D%5Cfrac%7B1%7D%7B2%7D)