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lim-x-0-sin-3x-tan-6x-




Question Number 198952 by ArifinTanjung last updated on 26/Oct/23
lim_(x→0)  ((sin 3x)/(tan 6x)) = ....?
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{3x}}{\mathrm{tan}\:\mathrm{6x}}\:=\:….? \\ $$
Answered by Mathspace last updated on 26/Oct/23
lim=(3/6)=(1/2)  (sin(3x)∼3x and tan(6x)∼6x)
$${lim}=\frac{\mathrm{3}}{\mathrm{6}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left({sin}\left(\mathrm{3}{x}\right)\sim\mathrm{3}{x}\:{and}\:{tan}\left(\mathrm{6}{x}\right)\sim\mathrm{6}{x}\right) \\ $$
Answered by Rasheed.Sindhi last updated on 26/Oct/23
lim_(x→0)  ((sin 3x)/(tan 6x)) = ....?  lim_(x→0)  ((sin 3x)/(  ((sin 6x)/(cos 6x))  )) =lim_(x→0)  ((sin 3x cos 6x)/(sin 6x))  =lim_(x→0) (( 3∙((sin 3x)/(3x))∙cos 6x)/(6∙((sin 6x)/(6x))))  =(1/2)∙(((lim_(3x→0)  ((sin 3x)/(3x)))(lim_(x→0)  cos 6x))/(lim_(6x→0) ((sin 6x)/(6x))))  =(1/2)∙(((1)(lim_(x→0)  cos 6x))/1)  =(1/2)cos 6(0)=(1/2)∙cos 0=(1/2)(1)=(1/2)
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{3x}}{\mathrm{tan}\:\mathrm{6x}}\:=\:….? \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{3x}}{\:\:\frac{\mathrm{sin}\:\mathrm{6}{x}}{\mathrm{cos}\:\mathrm{6}{x}}\:\:}\:=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{3x}\:\mathrm{cos}\:\mathrm{6}{x}}{\mathrm{sin}\:\mathrm{6}{x}} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\:\mathrm{3}\centerdot\frac{\mathrm{sin}\:\mathrm{3}{x}}{\mathrm{3}{x}}\centerdot\mathrm{cos}\:\mathrm{6}{x}}{\mathrm{6}\centerdot\frac{\mathrm{sin}\:\mathrm{6}{x}}{\mathrm{6}{x}}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\centerdot\frac{\left(\underset{\mathrm{3}{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{3}{x}}{\mathrm{3}{x}}\right)\left(\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{cos}\:\mathrm{6}{x}\right)}{\underset{\mathrm{6}{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:\mathrm{6}{x}}{\mathrm{6}{x}}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\centerdot\frac{\left(\mathrm{1}\right)\left(\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{cos}\:\mathrm{6}{x}\right)}{\mathrm{1}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{6}\left(\mathrm{0}\right)=\frac{\mathrm{1}}{\mathrm{2}}\centerdot\mathrm{cos}\:\mathrm{0}=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}\right)=\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Answered by essaad last updated on 26/Oct/23
=lim_(x→0) ((sin 3x)/(3x))×((6x)/(tan 6x))×(3/6)  =1×1×(3/6)  =(1/2)
$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sin}\:\mathrm{3}{x}}{\mathrm{3}{x}}×\frac{\mathrm{6}{x}}{{tan}\:\mathrm{6}{x}}×\frac{\mathrm{3}}{\mathrm{6}} \\ $$$$=\mathrm{1}×\mathrm{1}×\frac{\mathrm{3}}{\mathrm{6}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\: \\ $$

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