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




Question Number 122835 by liberty last updated on 20/Nov/20
 lim_(x→0) (1/x) ((1/(sin x)) − (3/(sin 3x)) ) =?
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\:\left(\frac{\mathrm{1}}{\mathrm{sin}\:{x}}\:−\:\frac{\mathrm{3}}{\mathrm{sin}\:\mathrm{3}{x}}\:\right)\:=? \\ $$
Answered by $@y@m last updated on 20/Nov/20
 lim_(x→0) (1/x) (((sin 3x−3sin x)/(sin x.sin 3x)) )    lim_(x→0) (1/x) (((−4sin^3 x)/(sin x.sin 3x)) )    lim_(x→0) (1/x) (((−4sin^2 x)/(sin 3x)) )    (−4)lim_(x→0) ((sin^2 x)/x^2 ) ÷ (lim_(x→0)   ((sin 3x)/(3x)) ×3)   =((−4)/3)
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\:\left(\frac{\mathrm{sin}\:\mathrm{3}{x}−\mathrm{3sin}\:{x}}{\mathrm{sin}\:{x}.\mathrm{sin}\:\mathrm{3}{x}}\:\right)\: \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\:\left(\frac{−\mathrm{4sin}\:^{\mathrm{3}} {x}}{\mathrm{sin}\:{x}.\mathrm{sin}\:\mathrm{3}{x}}\:\right)\: \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\:\left(\frac{−\mathrm{4sin}^{\mathrm{2}} {x}}{\mathrm{sin}\:\mathrm{3}{x}}\:\right)\: \\ $$$$\:\left(−\mathrm{4}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}^{\mathrm{2}} {x}}{{x}^{\mathrm{2}} }\:\boldsymbol{\div}\:\left(\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{sin}\:\mathrm{3}{x}}{\mathrm{3}{x}}\:×\mathrm{3}\right)\: \\ $$$$=\frac{−\mathrm{4}}{\mathrm{3}} \\ $$

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