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Question-200304




Question Number 200304 by Calculusboy last updated on 16/Nov/23
Answered by Sutrisno last updated on 17/Nov/23
lim_(n→∞) 0.2^(log_(√5) (((1/4)/(1−(1/2)))))   lim_(n→∞) 0.2^(log_(√5) ((1/3)))   lim_(n→∞) 0.2^(log_(0.2) ((1/3))^2 )   =(1/9)
$${lim}_{{n}\rightarrow\infty} \mathrm{0}.\mathrm{2}^{{log}_{\sqrt{\mathrm{5}}} \left(\frac{\frac{\mathrm{1}}{\mathrm{4}}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}}\right)} \\ $$$${lim}_{{n}\rightarrow\infty} \mathrm{0}.\mathrm{2}^{{log}_{\sqrt{\mathrm{5}}} \left(\frac{\mathrm{1}}{\mathrm{3}}\right)} \\ $$$${lim}_{{n}\rightarrow\infty} \mathrm{0}.\mathrm{2}^{{log}_{\mathrm{0}.\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{9}} \\ $$
Commented by Calculusboy last updated on 28/Nov/23
thanks sir
$$\boldsymbol{{thanks}}\:\boldsymbol{{sir}} \\ $$
Answered by MathematicalUser2357 last updated on 05/Jan/24
=lim_(n→∞) 0.2^(log_(√5) (1/2))   =lim_(n→∞) 0.2^(−(1/2)log_(1/5) (1/2))   =((√5)/2)
$$=\underset{{n}\rightarrow\infty} {\mathrm{lim}0}.\mathrm{2}^{\mathrm{log}_{\sqrt{\mathrm{5}}} \frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$=\underset{{n}\rightarrow\infty} {\mathrm{lim}0}.\mathrm{2}^{−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{log}_{\frac{\mathrm{1}}{\mathrm{5}}} \frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$=\frac{\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$

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