Trigonometrijski limes: smena x − pi/3 = u

TEKST ZADATKA

Odrediti graničnu vrednost:

\lim_{{x} \to {\frac {\pi} 3}} \frac {\sin{(x-\frac {\pi} 3)}} {1-2\cos{x}}

REŠENJE ZADATKA

Uvesti smenu:

x-\frac {\pi} 3=u \implies x=u+\frac {\pi} 3

Uvrstiti smenu u izraz:

\lim_{{u} \to {0}} \frac {\sin{u}} {1-2\cos{(u+\frac {\pi} 3)}}

Primeniti adicionu formulu: $\cos{(\alpha \pm \beta)}=\cos{\alpha}*\cos{\beta} \mp \sin{\alpha}*\sin{\beta}$

\lim_{{u} \to {0}} \frac {\sin{u}} {1-2(\cos{u}*\cos{\frac {\pi} 3} - \sin{u}*\sin{\frac {\pi} 3)}}

Uvrstiti vrednosti trigonometrijskih funkcija:

\lim_{{u} \to {0}} \frac {\sin{u}} {1-2(\frac 1 2\cos{u} - \frac {\sqrt{3}} 2\sin{u})}

Osloboditi se zagrade:

\lim_{{u} \to {0}} \frac {\sin{u}} {1-2*\frac 1 2\cos{u} +2* \frac {\sqrt{3}} 2\sin{u}} = \lim_{{u} \to {0}} \frac {\sin{u}} {1-\cancel{2}*\frac 1 {\cancel{2}}\cos{u} +\cancel{2}* \frac {\sqrt{3}} {\cancel{2}}\sin{u}} = \lim_{{u} \to {0}} \frac {\sin{u}} {1-\cos{u} +\sqrt{3}\sin{u}}

Primeniti formulu za sinus dvostrukog ugla: $\sin{(2\alpha)}=2\sin{\alpha}\cos{\alpha}$

\lim_{{u} \to {0}} \frac {2\sin{\frac u 2}\cos{\frac u 2}} {1-\cos{u} +\sqrt{3}*{2\sin{\frac u 2}\cos{\frac u 2}}}

Primeniti formulu za kosinus dvostrukog ugla: $\cos{2\alpha}=1-2\sin^2{\alpha}$

\lim_{{u} \to {0}} \frac {2\sin{\frac u 2}\cos{\frac u 2}} {1-(1-2\sin^2{\frac u 2}) +2\sqrt{3}{\sin{\frac u 2}\cos{\frac u 2}}}

Srediti izraz u imeniocu:

\lim_{{u} \to {0}} \frac {2\sin{\frac u 2}\cos{\frac u 2}} {1-1+2\sin^2{\frac u 2} +2\sqrt{3}{\sin{\frac u 2}\cos{\frac u 2}}} =\lim_{{u} \to {0}} \frac {2\sin{\frac u 2}\cos{\frac u 2}} {2\sin^2{\frac u 2} +2\sqrt{3}{\sin{\frac u 2}\cos{\frac u 2}}}

Izvući zajedničke činioce ispred zagrade:

\lim_{{u} \to {0}} \frac {2\sin{\frac u 2}\cos{\frac u 2}} {2\sin{\frac u 2} (\sin{\frac u 2}+\sqrt{3}\cos{\frac u 2})}

Skratiti zajedničke činioce:

\lim_{{u} \to {0}} \frac {\cancel{2\sin{\frac u 2}}\cos{\frac u 2}} {\cancel{2\sin{\frac u 2}} (\sin{\frac u 2}+\sqrt{3}\cos{\frac u 2})} =\lim_{{u} \to {0}} \frac {\cos{\frac u 2}} {\sin{\frac u 2}+\sqrt{3}\cos{\frac u 2}}

Zameniti $u=0.$

\frac {\cos{\frac 0 2}} {\sin{\frac 0 2}+\sqrt{3}\cos{\frac 0 2}}= \frac {\cos0} {\sin0+\sqrt{3}\cos0}

Uvrstiti vrednosti trigonometrijskih funkcija:

\frac 1 {0+\sqrt{3}*1}=\frac 1 {\sqrt{3}}=\frac {\sqrt{3}} 3

1. Trigonometrijski limes