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Question Number 157123 by cortano last updated on 20/Oct/21

lim_(x→0) ((log _(sin x) (cos x))/(log _(sin ((x/2))) (cos (x/2))))=?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{log}\:_{\mathrm{sin}\:{x}} \left(\mathrm{cos}\:{x}\right)}{\mathrm{log}\:_{\mathrm{sin}\:\left(\frac{{x}}{\mathrm{2}}\right)} \left(\mathrm{cos}\:\frac{{x}}{\mathrm{2}}\right)}=? \\ $$

Commented by john_santu last updated on 21/Oct/21

L=lim_(x→0) ((ln (cos x) ln (sin (x/2)))/(ln (cos (x/2)).ln (sin x))) L=lim_(x→0) (((ln (cos x))/(ln (cos (x/2))))).lim_(x→0) (((ln (sin (x/2)))/(ln (sin x)))) L=lim_(y→0) (((ln (cos 2y))/(ln (cos y)))).lim_(y→0) (((ln (sin y))/(ln (sin 2y)))) L=lim_(y→0) ((((−2sin 2y)/(cos 2y))/((−sin y)/(cos y)))).lim_(y→0) ((((cos y)/(sin y))/((2cos 2y)/(sin 2y)))) L=lim_(y→0) (((2tan 2y)/(tan y))).lim_(y→0) (((cot y)/(2cot 2y))) L=4×lim_(y→0) (((tan 2y)/(2tan y)))=4×1=4

$${L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left(\mathrm{cos}\:{x}\right)\:\mathrm{ln}\:\left(\mathrm{sin}\:\frac{{x}}{\mathrm{2}}\right)}{\mathrm{ln}\:\left(\mathrm{cos}\:\frac{{x}}{\mathrm{2}}\right).\mathrm{ln}\:\left(\mathrm{sin}\:{x}\right)} \\ $$$${L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{ln}\:\left(\mathrm{cos}\:{x}\right)}{\mathrm{ln}\:\left(\mathrm{cos}\:\frac{{x}}{\mathrm{2}}\right)}\right).\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{ln}\:\left(\mathrm{sin}\:\frac{{x}}{\mathrm{2}}\right)}{\mathrm{ln}\:\left(\mathrm{sin}\:{x}\right)}\right) \\ $$$${L}=\underset{{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{ln}\:\left(\mathrm{cos}\:\mathrm{2}{y}\right)}{\mathrm{ln}\:\left(\mathrm{cos}\:{y}\right)}\right).\underset{{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{ln}\:\left(\mathrm{sin}\:{y}\right)}{\mathrm{ln}\:\left(\mathrm{sin}\:\mathrm{2}{y}\right)}\right) \\ $$$${L}=\underset{{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\frac{−\mathrm{2sin}\:\mathrm{2}{y}}{\mathrm{cos}\:\mathrm{2}{y}}}{\frac{−\mathrm{sin}\:{y}}{\mathrm{cos}\:{y}}}\right).\underset{{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\frac{\mathrm{cos}\:{y}}{\mathrm{sin}\:{y}}}{\frac{\mathrm{2cos}\:\mathrm{2}{y}}{\mathrm{sin}\:\mathrm{2}{y}}}\right) \\ $$$${L}=\underset{{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{2tan}\:\mathrm{2}{y}}{\mathrm{tan}\:{y}}\right).\underset{{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{cot}\:{y}}{\mathrm{2cot}\:\mathrm{2}{y}}\right) \\ $$$${L}=\mathrm{4}×\underset{{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{tan}\:\mathrm{2}{y}}{\mathrm{2tan}\:{y}}\right)=\mathrm{4}×\mathrm{1}=\mathrm{4} \\ $$

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