mathop {lim }limits_{theta to 0} frac{{1 - co | vedclass

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Question

(C) Using the trigonometric identity $1 - \cos 	heta = 2 \sin^2(	heta/2)$,the limit becomes:$\mathop {\lim }\limits_{	heta 	o 0} \frac{2 \sin^2(

Vedclass · Accepted Answer

$\frac{1}{2}$

Vedclass · Answer

(C) Using the trigonometric identity $1 - \cos 	heta = 2 \sin^2(	heta/2)$,the limit becomes:$\mathop {\lim }\limits_{	heta 	o 0} \frac{2 \sin^2(	heta/2)}{	heta^2} = \mathop {\lim }\limits_{	heta 	o 0} \frac{2 \sin^2(	heta/2)}{4(	heta/2)^2} = \frac{2}{4} \mathop {\lim }\limits_{	heta 	o 0} \left( \frac{\sin(	heta/2)}{	heta/2} ight)^2 = \frac{1}{2} 	imes (1)^2 = \frac{1}{2}$.Alternatively,applying $L$-Hospital's rule:$\mathop {\lim }\limits_{	heta 	o 0} \frac{1 - \cos 	heta}{	heta^2} = \mathop {\lim }\limits_{	heta 	o 0} \frac{\sin 	heta}{2	heta} = \frac{1}{2} \mathop {\lim }\limits_{	heta 	o 0} \frac{\sin 	heta}{	heta} = \frac{1}{2} 	imes 1 = \frac{1}{2}$.

$\mathop {\lim }\limits_{\theta \to 0} \frac{{1 - \cos \theta }}{{{\theta ^2}}} = $

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