lim-x-0-3x-sin2x-2x-sin3x-

Question Number 2265 by B744237509 last updated on 12/Nov/15

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{3}{x}+{sin}\mathrm{2}{x}}{\mathrm{2}{x}+{sin}\mathrm{3}{x}}=? \\ $$$$ \\ $$$$ \\ $$

Answered by Filup last updated on 12/Nov/15

$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{3}{x}}{\mathrm{2}{x}+\mathrm{sin}\left(\mathrm{3}{x}\right)}+\frac{\mathrm{sin}\left(\mathrm{2}{x}\right)}{\mathrm{2}{x}+\mathrm{sin}\left(\mathrm{3}{x}\right)}\right) \\ $$$${Use}\:{L}'{H}\hat {{o}pital}'{s}\:{law} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{3}}{\mathrm{2}+\mathrm{3cos}\left(\mathrm{3}{x}\right)}+\frac{\mathrm{2cos}\left(\mathrm{2}{x}\right)}{\mathrm{2}+\mathrm{3cos}\left(\mathrm{3}{x}\right)}\right) \\ $$$$\mathrm{cos}\left(\mathrm{0}\right)=\mathrm{1} \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}+\mathrm{3cos}\left(\mathrm{3}×\mathrm{0}\right)}+\frac{\mathrm{2cos}\left(\mathrm{2}×\mathrm{0}\right)}{\mathrm{2}+\mathrm{3cos}\left(\mathrm{3}×\mathrm{0}\right)} \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}+\mathrm{3}}+\frac{\mathrm{2}}{\mathrm{2}+\mathrm{3}} \\ $$$$=\frac{\mathrm{3}}{\mathrm{5}}+\frac{\mathrm{2}}{\mathrm{5}} \\ $$$$=\mathrm{1} \\ $$$$\therefore\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{3}{x}+\mathrm{sin}\left(\mathrm{2}{x}\right)}{\mathrm{2}{x}+\mathrm{sin}\left(\mathrm{3}{x}\right)}=\mathrm{1} \\ $$

Commented by Filup last updated on 12/Nov/15

$$\mathrm{Expanding}\:\mathrm{the}\:\mathrm{fraction}\:\mathrm{into}\:\mathrm{two}\:\mathrm{parts} \\ $$$$\mathrm{isnt}\:\mathrm{nessisary}.\:\mathrm{You}\:\mathrm{can}\:\mathrm{start}\:\mathrm{on}\:\mathrm{line}\:\mathrm{2} \\ $$

Answered by Rasheed Soomro last updated on 12/Nov/15

$${A}\:{way}\:{which}\:{doesn}'{t}\:{use}\:{derivative}. \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{3}{x}+{sin}\mathrm{2}{x}}{\mathrm{2}{x}+{sin}\mathrm{3}{x}} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\frac{\mathrm{3}{x}+{sin}\mathrm{2}{x}}{\mathrm{2}{x}}×\mathrm{2}{x}}{\frac{\mathrm{2}{x}+{sin}\mathrm{3}{x}}{\mathrm{3}{x}}×\mathrm{3}{x}} \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}}\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{\frac{\mathrm{3}{x}}{\mathrm{2}{x}}+\frac{{sin}\:\mathrm{2}{x}}{\mathrm{2}{x}}}{\frac{\mathrm{2}{x}}{\mathrm{3}{x}}+\frac{{sin}\:\mathrm{3}{x}}{\mathrm{3}{x}}} \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}}\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{\frac{\mathrm{3}}{\mathrm{2}}+\frac{{sin}\:\mathrm{2}{x}}{\mathrm{2}{x}}}{\frac{\mathrm{2}}{\mathrm{3}}+\frac{{sin}\:\mathrm{3}{x}}{\mathrm{3}{x}}} \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}}×\frac{\frac{\mathrm{3}}{\mathrm{2}}+\underset{\mathrm{2}{x}\rightarrow\mathrm{0}} {{lim}}\frac{{sin}\:\mathrm{2}{x}}{\mathrm{2}{x}}}{\frac{\mathrm{2}}{\mathrm{3}}+\underset{\mathrm{3}{x}\rightarrow\mathrm{0}} {{lim}}\frac{{sin}\:\mathrm{3}{x}}{\mathrm{3}{x}}}\:\:\:\:\:\left[{x}\rightarrow\mathrm{0\begin{cases}{\mathrm{2}{x}\rightarrow\mathrm{0}}\\{\mathrm{3}{x}\rightarrow\mathrm{0}}\end{cases}}\:\:\right] \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}}×\frac{\frac{\mathrm{3}}{\mathrm{2}}+\mathrm{1}}{\frac{\mathrm{2}}{\mathrm{3}}+\mathrm{1}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{{sin}\:{x}}{{x}}=\mathrm{1}\right] \\ $$$$\:=\frac{\mathrm{2}}{\mathrm{3}}×\frac{\mathrm{5}}{\mathrm{2}}×\frac{\mathrm{3}}{\mathrm{5}}=\mathrm{1} \\ $$

Commented by Filup last updated on 12/Nov/15

$$\mathrm{I}\:\mathrm{knew}\:\mathrm{there}\:\mathrm{was}\:\mathrm{a}\:\mathrm{way}\:\mathrm{but}\:\mathrm{i}\:\mathrm{didnt}\:\mathrm{know} \\ $$$$\mathrm{how}\:\mathrm{to}\:\mathrm{do}\:\mathrm{it}.\:\mathrm{Nice}! \\ $$

Commented by B744237509 last updated on 12/Nov/15

$${thanx} \\ $$

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