Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

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)

$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sin}\:\mathrm{3}{x}}{\mathrm{3}{x}}×\frac{\mathrm{6}{x}}{{tan}\:\mathrm{6}{x}}×\frac{\mathrm{3}}{\mathrm{6}} \\ $$$$=\mathrm{1}×\mathrm{1}×\frac{\mathrm{3}}{\mathrm{6}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\: \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com