Limits Questions

Question Number 102510 by Study last updated on 09/Jul/20

$${li}\underset{\bigtriangleup{x}\rightarrow\mathrm{0}} {{m}}\frac{{sin}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{3}}{x}+\frac{\mathrm{1}}{\mathrm{3}}\bigtriangleup{x}\right)−{sin}^{\mathrm{2}} \frac{\mathrm{1}}{\mathrm{3}}{x}}{\bigtriangleup{x}}=? \\$$

Answered by bemath last updated on 09/Jul/20

$$\underset{\Delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:^{\mathrm{2}} \frac{\mathrm{1}}{\mathrm{3}}\left({x}+\Delta{x}\right)−\mathrm{sin}\:^{\mathrm{2}} \frac{\mathrm{1}}{\mathrm{3}}{x}}{\Delta{x}}\:= \\$$$$\frac{{d}}{{dx}}\:\left[\:\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{3}}{x}\right)\right]\:=\:\frac{\mathrm{1}}{\mathrm{3}}\mathrm{sin}\:\left(\frac{\mathrm{2}{x}}{\mathrm{3}}\right)\: \\$$

Answered by Dwaipayan Shikari last updated on 09/Jul/20

$${Method}\:\mathrm{1} \\$$$$\frac{\mathrm{2}}{\mathrm{3}}{sin}\frac{{x}}{\mathrm{3}}{cos}\frac{{x}}{\mathrm{3}}=\frac{\mathrm{1}}{\mathrm{3}}{sin}\frac{\mathrm{2}{x}}{\mathrm{3}} \\$$$${Method}\:\mathrm{2} \\$$$$\underset{\bigtriangleup{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left({sin}\left(\frac{{x}}{\mathrm{3}}+\bigtriangleup\frac{{x}}{\mathrm{3}}\right)+{sin}\frac{{x}}{\mathrm{3}}\right)\left({sin}\left(\frac{{x}}{\mathrm{3}}+\bigtriangleup\frac{{x}}{\mathrm{3}}\right)−{sin}\frac{{x}}{\mathrm{3}}\right)}{\bigtriangleup{x}} \\$$$$\\$$$$\underset{\bigtriangleup{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\mathrm{2}{sin}\left(\frac{\frac{\mathrm{2}{x}}{\mathrm{3}}+\bigtriangleup\frac{{x}}{\mathrm{3}}}{\mathrm{2}}\right){cos}\bigtriangleup\frac{{x}}{\mathrm{6}}\right)\left(\mathrm{2}{cos}\left(\frac{\frac{\mathrm{2}{x}}{\mathrm{3}}+\bigtriangleup\frac{{x}}{\mathrm{3}}}{\mathrm{2}}\right){sin}\bigtriangleup\frac{{x}}{\mathrm{6}}\right)}{\bigtriangleup{x}} \\$$$$\mathrm{2}{sin}\frac{\mathrm{2}{x}+\bigtriangleup{x}}{\mathrm{3}}\:.\frac{\frac{\bigtriangleup{x}}{\mathrm{6}}}{\bigtriangleup{x}}=\frac{\mathrm{1}}{\mathrm{3}}{sin}\frac{\mathrm{2}{x}}{\mathrm{3}} \\$$$$\\$$