184.

Trigonometrijski limes

TEKST ZADATKA

Odrediti graničnu vrednost:

lim⁡x→π3sin⁡(x−π3)1−2cos⁡x\lim_{{x} \to {\frac {\pi} 3}} \frac {\sin{(x-\frac {\pi} 3)}} {1-2\cos{x}}
REŠENJE ZADATKA

Uvesti smenu:

x−π3=u  ⟹  x=u+π3x-\frac {\pi} 3=u \implies x=u+\frac {\pi} 3
DODATNO OBJAŠNJENJE

Uvrstiti smenu u izraz:

lim⁡u→0sin⁡u1−2cos⁡(u+π3)\lim_{{u} \to {0}} \frac {\sin{u}} {1-2\cos{(u+\frac {\pi} 3)}}

Primeniti adicionu formulu: cos⁡(α±β)=cos⁡α∗cos⁡β∓sin⁡α∗sin⁡β \cos{(\alpha \pm \beta)}=\cos{\alpha}*\cos{\beta} \mp \sin{\alpha}*\sin{\beta}

lim⁡u→0sin⁡u1−2(cos⁡u∗cos⁡π3−sin⁡u∗sin⁡π3)\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→0sin⁡u1−2(12cos⁡u−32sin⁡u)\lim_{{u} \to {0}} \frac {\sin{u}} {1-2(\frac 1 2\cos{u} - \frac {\sqrt{3}} 2\sin{u})}
DODATNO OBJAŠNJENJE

Osloboditi se zagrade:

lim⁡u→0sin⁡u1−2∗12cos⁡u+2∗32sin⁡u=lim⁡u→0sin⁡u1−2∗12cos⁡u+2∗32sin⁡u=lim⁡u→0sin⁡u1−cos⁡u+3sin⁡u\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⁡(2α)=2sin⁡αcos⁡α\sin{(2\alpha)}=2\sin{\alpha}\cos{\alpha}

lim⁡u→02sin⁡u2cos⁡u21−cos⁡u+3∗2sin⁡u2cos⁡u2\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}}}
DODATNO OBJAŠNJENJE

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

lim⁡u→02sin⁡u2cos⁡u21−(1−2sin⁡2u2)+23sin⁡u2cos⁡u2\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}}}
DODATNO OBJAŠNJENJE

Srediti izraz u imeniocu:

lim⁡u→02sin⁡u2cos⁡u21−1+2sin⁡2u2+23sin⁡u2cos⁡u2=lim⁡u→02sin⁡u2cos⁡u22sin⁡2u2+23sin⁡u2cos⁡u2\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→02sin⁡u2cos⁡u22sin⁡u2(sin⁡u2+3cos⁡u2)\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→02sin⁡u2cos⁡u22sin⁡u2(sin⁡u2+3cos⁡u2)=lim⁡u→0cos⁡u2sin⁡u2+3cos⁡u2\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.u=0.

cos⁡02sin⁡02+3cos⁡02=cos⁡0sin⁡0+3cos⁡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:

10+3∗1=13=33\frac 1 {0+\sqrt{3}*1}=\frac 1 {\sqrt{3}}=\frac {\sqrt{3}} 3
DODATNO OBJAŠNJENJE

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