mathop {lim }limits_{x to 0} frac{{sin left( | vedclass

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Question

(D) આપણે લક્ષની કિંમત મેળવવાની છે: $\mathop {\lim }\limits_{x 	o 0} \frac{{\sin \left( {\pi {{\cos }^2}x} ight)}}{{{x^2}}}$નિત્યસમ $\cos^2 x = 1 -

Vedclass · Accepted Answer

$\pi$

Vedclass · Answer

(D) આપણે લક્ષની કિંમત મેળવવાની છે: $\mathop {\lim }\limits_{x 	o 0} \frac{{\sin \left( {\pi {{\cos }^2}x} ight)}}{{{x^2}}}$નિત્યસમ $\cos^2 x = 1 - \sin^2 x$ નો ઉપયોગ કરતા:$= \mathop {\lim }\limits_{x 	o 0} \frac{{\sin \left( {\pi (1 - \sin^2 x)} ight)}}{{{x^2}}}$$= \mathop {\lim }\limits_{x 	o 0} \frac{{\sin \left( {\pi - \pi \sin^2 x} ight)}}{{{x^2}}}$$\sin(\pi - 	heta) = \sin 	heta$ હોવાથી:$= \mathop {\lim }\limits_{x 	o 0} \frac{{\sin \left( {\pi \sin^2 x} ight)}}{{{x^2}}}$$\pi \sin^2 x$ વડે ગુણતા અને ભાગતા:$= \mathop {\lim }\limits_{x 	o 0} \left( \frac{{\sin \left( {\pi \sin^2 x} ight)}}{{\pi \sin^2 x}} 	imes \frac{{\pi \sin^2 x}}{{{x^2}}} ight)$પ્રમાણિત લક્ષ $\mathop {\lim }\limits_{	heta 	o 0} \frac{{\sin 	heta}}{	heta} = 1$ નો ઉપયોગ કરતા:$= 1 	imes \pi 	imes (\mathop {\lim }\limits_{x 	o 0} \frac{{\sin x}}{x})^2 = 1 	imes \pi 	imes 1^2 = \pi$

$\mathop {\lim }\limits_{x \to 0} \frac{{\sin \left( {\pi {{\cos }^2}x} \right)}}{{{x^2}}}$ ની કિંમત શોધો.

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