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Question Number 169508 by Giantyusuf last updated on 01/May/22
Evaluate if the limit exist  lim_(n→∞) (((3^n +(−2)^(n+1) )/(3^(n−2) −2^(2n−1) )))
$${Evaluate}\:{if}\:{the}\:{limit}\:{exist} \\ $$$${lim}_{\boldsymbol{{n}}\rightarrow\infty} \left(\frac{\mathrm{3}^{\boldsymbol{{n}}} +\left(−\mathrm{2}\right)^{\boldsymbol{{n}}+\mathrm{1}} }{\mathrm{3}^{\boldsymbol{{n}}−\mathrm{2}} −\mathrm{2}^{\mathrm{2}\boldsymbol{{n}}−\mathrm{1}} }\right) \\ $$
Answered by qaz last updated on 01/May/22
lim_(n→∞) ((3^n +(−2)^(n+1) )/(3^(n−2) −2^(2n−1) ))=lim_(n→∞) ((((3/4))^n −2(−(1/2))^n )/(3^(−2) ((3/4))^n −2^(−1) ))=0
$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{3}^{\mathrm{n}} +\left(−\mathrm{2}\right)^{\mathrm{n}+\mathrm{1}} }{\mathrm{3}^{\mathrm{n}−\mathrm{2}} −\mathrm{2}^{\mathrm{2n}−\mathrm{1}} }=\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{n}} −\mathrm{2}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{n}} }{\mathrm{3}^{−\mathrm{2}} \left(\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{n}} −\mathrm{2}^{−\mathrm{1}} }=\mathrm{0} \\ $$

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