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Question Number 154586 by Dadoubleny last updated on 19/Sep/21

Answered by ARUNG_Brandon_MBU last updated on 19/Sep/21

lim_(n→∞) ((n+1)^α −n^α )  lim_(n→∞) (n^α (1+(1/n))^α −n^α )  lim_(n→∞) (n^α (1+(α/n))−n^α )  lim_(n→∞) ((α/n))=0

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\left({n}+\mathrm{1}\right)^{\alpha} −{n}^{\alpha} \right) \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left({n}^{\alpha} \left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{\alpha} −{n}^{\alpha} \right) \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left({n}^{\alpha} \left(\mathrm{1}+\frac{\alpha}{{n}}\right)−{n}^{\alpha} \right) \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\alpha}{{n}}\right)=\mathrm{0} \\ $$

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