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x-lim-0-sin3x-2x-2-5x-1-




Question Number 76248 by john santuy last updated on 25/Dec/19
x^(lim) →0^(    (((sin3x)/(2x)))^(2/(5x+1)) ) = ?
$$\overset{{lim}} {{x}}\rightarrow\overset{\:\:\:\:\left(\frac{{sin}\mathrm{3}{x}}{\mathrm{2}{x}}\right)^{\frac{\mathrm{2}}{\mathrm{5}{x}+\mathrm{1}}} } {\mathrm{0}}=\:?\:\: \\ $$
Commented by Mikael_786 last updated on 25/Dec/19
lim_(x→0) ((sin3x)/(2x))=lim_(x→0) ((3sin3x)/(3.2x))=(3/2)lim_(x→0) ((sin3x)/(3x))=(3/2)  lim_(x→0) (2/(5x+1))=(2/(5×0+1))=2  lim_(x→0) (((sin3x)/(2x)))^(2/(5x+1)) =((3/2))^2 =(9/4)
$$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{{sin}\mathrm{3}{x}}{\mathrm{2}{x}}=\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{\mathrm{3}{sin}\mathrm{3}{x}}{\mathrm{3}.\mathrm{2}{x}}=\frac{\mathrm{3}}{\mathrm{2}}\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{{sin}\mathrm{3}{x}}{\mathrm{3}{x}}=\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{\mathrm{2}}{\mathrm{5}{x}+\mathrm{1}}=\frac{\mathrm{2}}{\mathrm{5}×\mathrm{0}+\mathrm{1}}=\mathrm{2} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\left(\frac{{sin}\mathrm{3}{x}}{\mathrm{2}{x}}\right)^{\frac{\mathrm{2}}{\mathrm{5}{x}+\mathrm{1}}} =\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} =\frac{\mathrm{9}}{\mathrm{4}} \\ $$
Commented by turbo msup by abdo last updated on 25/Dec/19
let f(x)=(((sin(3x))/(2x)))^(2/(5x+1))   ⇒f(x)=e^((2/(5x+1))ln(((sin(3x))/(2x))))    we have  ((sin(3x))/(2x)) ∼ ((3x)/(2x)) =(3/2) ⇒  lim_(x→0)  f(x)= e^(2ln((3/2)))   =((3/2))^2  =(9/4)
$${let}\:{f}\left({x}\right)=\left(\frac{{sin}\left(\mathrm{3}{x}\right)}{\mathrm{2}{x}}\right)^{\frac{\mathrm{2}}{\mathrm{5}{x}+\mathrm{1}}} \\ $$$$\Rightarrow{f}\left({x}\right)={e}^{\frac{\mathrm{2}}{\mathrm{5}{x}+\mathrm{1}}{ln}\left(\frac{{sin}\left(\mathrm{3}{x}\right)}{\mathrm{2}{x}}\right)} \:\:\:{we}\:{have} \\ $$$$\frac{{sin}\left(\mathrm{3}{x}\right)}{\mathrm{2}{x}}\:\sim\:\frac{\mathrm{3}{x}}{\mathrm{2}{x}}\:=\frac{\mathrm{3}}{\mathrm{2}}\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \:{f}\left({x}\right)=\:{e}^{\mathrm{2}{ln}\left(\frac{\mathrm{3}}{\mathrm{2}}\right)} \\ $$$$=\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} \:=\frac{\mathrm{9}}{\mathrm{4}} \\ $$
Commented by john santuy last updated on 26/Dec/19
thanks sir
$${thanks}\:{sir} \\ $$

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