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Zadatak 0838

Test: MKsIVBGI_073

Izračunaj graničnu vrednost niza: $$\lim \limits_{n \to +\infty }\left ( \sqrt{n^2+2n+2}-\sqrt{n^2-4n+3} \right )$$

Rešenje:

$$...=\lim \limits_{n \to +\infty }\left ( \sqrt{n^2+2n+2}-\sqrt{n^2-4n+3} \right )\cdot \frac{\left ( \sqrt{n^2+2n+2}+\sqrt{n^2-4n+3} \right )}{\left ( \sqrt{n^2+2n+2}+\sqrt{n^2-4n+3} \right )}$$

$$=\lim \limits_{n \to +\infty }\frac{n^2+2n+2-n^2+4n-3}{\left ( \sqrt{n^2+2n+2}+\sqrt{n^2-4n+3} \right )}$$

$$=\lim \limits_{n \to +\infty }\frac{n(6-\frac{1}{n})}{n(\sqrt{1+\frac{2}{n}+\frac{2}{n^2}}+\sqrt{1-\frac{4}{n}+\frac{3}{n^2}})}$$

$$=\frac{6}{2}=3$$

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