Trigonometrijski limes: 1 − cos x kroz x kvadrat

TEKST ZADATKA

Odrediti graničnu vrednost:

\lim_{{x} \to {0}}\frac{1-\cos x}{x^2}

REŠENJE ZADATKA

Primeniti formulu za sinus poluugla: $\sin^2{\frac {\alpha} 2}=\frac {1-\cos{\alpha}} 2$

\lim_{{x} \to {0}}\frac{2\sin^2\frac x2}{x^2}

Raščlaniti izraz.

\lim_{{x} \to {0}}2\cdot\lim_{{x} \to {0}}\frac{\sin\frac x2}{x}\cdot\lim_{{x} \to {0}}\frac{\sin\frac x2}{x}

Pomnožiti imenioce sa $\frac22.$

2\cdot\lim_{{x} \to {0}}\frac{\sin\frac x2}{x\cdot\frac22}\cdot\lim_{{x} \to {0}}\frac{\sin\frac x2}{x\cdot\frac22}\\ 2\cdot\lim_{{x} \to {0}}\frac{\sin\frac x2}{\frac x2}\cdot\lim_{{x} \to {0}}\frac12\cdot\lim_{{x} \to {0}}\frac{\sin\frac x2}{\frac x2}\cdot\lim_{{x} \to {0}}\frac12 \\ 2\cdot\lim_{{x} \to {0}}\frac{\sin\frac x2}{\frac x2}\cdot\frac12\cdot\lim_{{x} \to {0}}\frac{\sin\frac x2}{\frac x2}\cdot\frac12 \\ \frac12\lim_{{x} \to {0}}\frac{\sin\frac x2}{\frac x2}\cdot\lim_{{x} \to {0}}\frac{\sin\frac x2}{\frac x2}

Primeniti tablični limes: $\lim_{{x} \to {0}}\frac {\sin{x}} {x}=1$

\frac12\cdot1\cdot1 \\ \frac12

$\lim_{{x} \to {0}}\frac{1-\cos x}{x^2}$