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lim-x-0-1-x-1-sin-x-3-sin-3x-




Question Number 75553 by aliesam last updated on 12/Dec/19
lim_(x→0) (1/x)((1/(sin(x)))−(3/(sin(3x))))
$$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{\mathrm{1}}{{x}}\left(\frac{\mathrm{1}}{{sin}\left({x}\right)}−\frac{\mathrm{3}}{{sin}\left(\mathrm{3}{x}\right)}\right) \\ $$
Commented by mathmax by abdo last updated on 12/Dec/19
f(x)=((sin(3x)−3sinx)/(x sin(x)sin(3x)))  we have sinx=Σ_(n=0) ^∞  (((−1)^n x^(2n+1) )/((2n+1)!)) ⇒  sinx =x−(x^3 /(3!))+o(u^5 ) ⇒sin(3x)=3x−(((3x)^3 )/(3!)) +o(x^5 )  =3x−((27 x^3 )/(3.2)) +o(x^5 ) =3x−((9x^3 )/2) +o(x^5 )  ⇒  sin(3x)−3sinx ∼3x−((9x^3 )/2)−3x +(x^3 /2) +o(x^5 ) =−4x^3  +o(x^5 )  and xsin(x)sin(3x)∼3x^3  ⇒f(x) ∼−(4/3) +o(x^2 ) ⇒  lim_(x→0)  f(x)=((−4)/3)
$${f}\left({x}\right)=\frac{{sin}\left(\mathrm{3}{x}\right)−\mathrm{3}{sinx}}{{x}\:{sin}\left({x}\right){sin}\left(\mathrm{3}{x}\right)}\:\:{we}\:{have}\:{sinx}=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}\:\Rightarrow \\ $$$${sinx}\:={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+{o}\left({u}^{\mathrm{5}} \right)\:\Rightarrow{sin}\left(\mathrm{3}{x}\right)=\mathrm{3}{x}−\frac{\left(\mathrm{3}{x}\right)^{\mathrm{3}} }{\mathrm{3}!}\:+{o}\left({x}^{\mathrm{5}} \right) \\ $$$$=\mathrm{3}{x}−\frac{\mathrm{27}\:{x}^{\mathrm{3}} }{\mathrm{3}.\mathrm{2}}\:+{o}\left({x}^{\mathrm{5}} \right)\:=\mathrm{3}{x}−\frac{\mathrm{9}{x}^{\mathrm{3}} }{\mathrm{2}}\:+{o}\left({x}^{\mathrm{5}} \right)\:\:\Rightarrow \\ $$$${sin}\left(\mathrm{3}{x}\right)−\mathrm{3}{sinx}\:\sim\mathrm{3}{x}−\frac{\mathrm{9}{x}^{\mathrm{3}} }{\mathrm{2}}−\mathrm{3}{x}\:+\frac{{x}^{\mathrm{3}} }{\mathrm{2}}\:+{o}\left({x}^{\mathrm{5}} \right)\:=−\mathrm{4}{x}^{\mathrm{3}} \:+{o}\left({x}^{\mathrm{5}} \right) \\ $$$${and}\:{xsin}\left({x}\right){sin}\left(\mathrm{3}{x}\right)\sim\mathrm{3}{x}^{\mathrm{3}} \:\Rightarrow{f}\left({x}\right)\:\sim−\frac{\mathrm{4}}{\mathrm{3}}\:+{o}\left({x}^{\mathrm{2}} \right)\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \:{f}\left({x}\right)=\frac{−\mathrm{4}}{\mathrm{3}} \\ $$
Commented by aliesam last updated on 12/Dec/19
thank you sir
$${thank}\:{you}\:{sir}\: \\ $$
Commented by abdomathmax last updated on 13/Dec/19
you are welcome sir.
$${you}\:{are}\:{welcome}\:{sir}. \\ $$

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