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Question Number 169147 by mathlove last updated on 25/Apr/22

lim_(x→(π/3))  ((1−2cosx)/(sin(x−(π/3))))=?

$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{3}}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{2}{cosx}}{{sin}\left({x}−\frac{\pi}{\mathrm{3}}\right)}=? \\ $$

Commented by cortano1 last updated on 25/Apr/22

 let x−(π/3)=h   lim_(h→0)  ((1−2cos ((π/3)+h))/(sin h))  = lim_(h→0) ((2sin ((π/3)+h))/(cos h))=2×(1/2)(√3)=(√3)

$$\:{let}\:{x}−\frac{\pi}{\mathrm{3}}={h} \\ $$$$\:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{2cos}\:\left(\frac{\pi}{\mathrm{3}}+{h}\right)}{\mathrm{sin}\:{h}} \\ $$$$=\:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{2sin}\:\left(\frac{\pi}{\mathrm{3}}+{h}\right)}{\mathrm{cos}\:{h}}=\mathrm{2}×\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{3}}=\sqrt{\mathrm{3}} \\ $$

Commented by infinityaction last updated on 25/Apr/22

       p   =     lim_(x→(π/3))  2(((cos (π/3)−cos x)/(sin (x−(π/3)))))        p  =  lim_(x→(π/3)) 2(((2sin (((x+π/3)/2))sin( ((x−π/3)/2)))/(2sin (((x−π/3)/2))cos (((x−π/3)/2)))))       p     =   2sin (((2π)/3))       p   =    2×((√3)/2)   =     (√3)

$$\:\:\:\:\:\:\:{p}\:\:\:=\:\:\:\:\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{3}}} {\mathrm{lim}}\:\mathrm{2}\left(\frac{\mathrm{cos}\:\frac{\pi}{\mathrm{3}}−\mathrm{cos}\:{x}}{\mathrm{sin}\:\left({x}−\frac{\pi}{\mathrm{3}}\right)}\right) \\ $$$$\:\:\:\:\:\:{p}\:\:=\:\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{3}}} {\mathrm{lim}2}\left(\frac{\mathrm{2sin}\:\left(\frac{{x}+\pi/\mathrm{3}}{\mathrm{2}}\right)\mathrm{sin}\left(\:\frac{{x}−\pi/\mathrm{3}}{\mathrm{2}}\right)}{\mathrm{2sin}\:\left(\frac{{x}−\pi/\mathrm{3}}{\mathrm{2}}\right)\mathrm{cos}\:\left(\frac{{x}−\pi/\mathrm{3}}{\mathrm{2}}\right)}\right) \\ $$$$\:\:\:\:\:{p}\:\:\:\:\:=\:\:\:\mathrm{2sin}\:\left(\frac{\mathrm{2}\pi}{\mathrm{3}}\right) \\ $$$$\:\:\:\:\:{p}\:\:\:=\:\:\:\:\mathrm{2}×\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:\:\:=\:\:\:\:\:\sqrt{\mathrm{3}} \\ $$$$ \\ $$

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