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Question Number 201332 by Rodier97 last updated on 04/Dec/23
          calcul :        1.    lim_(n→+∞ ) ((sin(n^2 )−cos(n^3 ))/n)          2.     lim_(n→+∞) ^n (√(3−sin(n^2 )))         3.     lim_(n→+∞)   (n/(n+1)) e^(i((nπ)/3))
$$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:{calcul}\:: \\ $$$$ \\ $$$$\:\:\:\:\mathrm{1}.\:\:\:\:{lim}_{{n}\rightarrow+\infty\:} \frac{{sin}\left({n}^{\mathrm{2}} \right)−{cos}\left({n}^{\mathrm{3}} \right)}{{n}}\: \\ $$$$ \\ $$$$\:\:\:\:\:\mathrm{2}.\:\:\:\:\:{lim}_{{n}\rightarrow+\infty} \:^{{n}} \sqrt{\mathrm{3}−{sin}\left({n}^{\mathrm{2}} \right)} \\ $$$$ \\ $$$$\:\:\:\:\:\mathrm{3}.\:\:\:\:\:{lim}_{{n}\rightarrow+\infty} \:\:\frac{{n}}{{n}+\mathrm{1}}\:{e}^{{i}\frac{{n}\pi}{\mathrm{3}}} \: \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Answered by MM42 last updated on 04/Dec/23
1)0      2)1      3)  (n/(n+1))(cos((nπ)/3)+isin((nπ)/3))  n=3k⇒coskπ+isinkπ { ((k=2m→1)),((k=2m+1→−1)) :}  ⇒lim ; no exist
$$\left.\mathrm{1}\left.\right)\left.\mathrm{0}\:\:\:\:\:\:\mathrm{2}\right)\mathrm{1}\:\:\:\:\:\:\mathrm{3}\right) \\ $$$$\frac{{n}}{{n}+\mathrm{1}}\left({cos}\frac{{n}\pi}{\mathrm{3}}+{isin}\frac{{n}\pi}{\mathrm{3}}\right) \\ $$$${n}=\mathrm{3}{k}\Rightarrow{cosk}\pi+{isink}\pi\begin{cases}{{k}=\mathrm{2}{m}\rightarrow\mathrm{1}}\\{{k}=\mathrm{2}{m}+\mathrm{1}\rightarrow−\mathrm{1}}\end{cases} \\ $$$$\Rightarrow{lim}\:;\:{no}\:{exist} \\ $$

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