mathop {Lim}limits_{x to 0} frac{{log _{{{sin | vedclass

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(C) ધારો કે $L = \mathop {Lim}\limits_{x 	o 0} \frac{{\log _{{{\sin }^2}x}}\cos x}{{\log _{{{\sin }^2}\frac{x}{2}}}\cos \frac{x}{2}}$.બેઝ ચેન્જ ફોર્મ

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(C) ધારો કે $L = \mathop {Lim}\limits_{x 	o 0} \frac{{\log _{{{\sin }^2}x}}\cos x}{{\log _{{{\sin }^2}\frac{x}{2}}}\cos \frac{x}{2}}$.બેઝ ચેન્જ ફોર્મ્યુલા $\log_a b = \frac{\ln b}{\ln a}$ નો ઉપયોગ કરતા:$L = \mathop {Lim}\limits_{x 	o 0} \frac{\ln(\cos x) / \ln(\sin^2 x)}{\ln(\cos(x/2)) / \ln(\sin^2(x/2))}$.જ્યારે $x 	o 0$,$\cos x \approx 1 - x^2/2$ અને $\sin x \approx x$,તેથી $\ln(\cos x) \approx -x^2/2$ અને $\ln(\sin^2 x) \approx 2\ln x$.તે જ રીતે,$\ln(\cos(x/2)) \approx -x^2/8$ અને $\ln(\sin^2(x/2)) \approx 2\ln x$.આ કિંમતો મૂકતા:$L = \mathop {Lim}\limits_{x 	o 0} \frac{-x^2/2}{2\ln x} \cdot \frac{2\ln x}{-x^2/8} = \frac{8}{2} = 4$.

$\mathop {Lim}\limits_{x \to 0} \frac{{\log _{{{\sin }^2}x}}\cos x}{{\log _{{{\sin }^2}\frac{x}{2}}}\cos \frac{x}{2}}$ ની કિંમત કેટલી થાય?

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