Question

${lim}_{x\to a}{\left\{\frac{\sin x}{\sin a}\right\}}^{\frac{1}{x-a}}$

Answer 1

${lim}_{x\to a}{\left\{\frac{\sin x}{\sin a}\right\}}^{\frac{1}{x-a}}={lim}_{x\to a}{\{1+(\frac{\sin x}{\sin a}-1)\}}^{\frac{1}{x-a}}=e^{{lim}_{x\to a}\left\{\frac{(\frac{\sin x}{\sin a}-1)}{x-a}\right\}}=e^{{lim}_{x\to a}\left\{\frac{\left(\frac{\sin x-\sin a}{\sin a}\right)}{x-a}\right\}}=e^{{lim}_{x\to a}\left\{\left(\frac{\sin x-\sin a}{(x-a)\sin a}\right)\right\}}=e^{{lim}_{x\to a}\left\{\left(\frac{2cos\left(\frac{x+a}{2}\right)\sin \left(\frac{x-a}{2}\right)}{(x-a)\sin a}\right)\right\}}=e^{{lim}_{x\to a}\left\{\left(\frac{2\cos \left(\frac{x+a}{2}\right)}{\sin a}\right)\right\}lim\left\{\left(\frac{\sin \left(\frac{x-a}{2}\right)}{2\left(\frac{x-a}{2}\right)}\right)\right\}}=e^{\frac{2\cos a}{2\sin a}}=e^{\cot a}$