$\mathrm{f}\left(\mathrm{x}\right) = {\left(\frac{\sin \mathrm{x}}{\sin \mathrm{a}}\right)}^{\frac{1}{\mathrm{x} -\mathrm{a}}} \mathrm{if} \mathrm{x}\ne \mathrm{a}$ and f(x) = k if x = a and f(x) is continuous at x = a then k =

$\begin{array}{rcl}\underset{\mathrm{x}\to \mathrm{a}}{\lim }\mathrm{f}\left(\mathrm{x}\right) & =& \underset{x\to a}{\lim } {\left(\frac{\sin \mathrm{x}}{\sin \mathrm{a}}\right)}^{\frac{1}{\mathrm{x} -\mathrm{a}}}\\ & =& e^{\underset{x\to a}{\lim } \left(\frac{\sin \mathrm{x}}{\sin \mathrm{a}}-1\right)\frac{1}{\mathrm{x} -\mathrm{a}}}\\ & =& e^{\underset{x\to a}{\lim } \left(\frac{\sin \mathrm{x}-\mathrm{sina}}{\mathrm{x} -\mathrm{a}}\right)\frac{1}{\sin \mathrm{a}}}\\ & =& {\mathrm{e}}^{\underset{\mathrm{x}\to \mathrm{a}}{\lim } \left(\frac{\cos \mathrm{x}-0}{1}\right)\frac{1}{\sin \mathrm{a}}}\\ & =& {\mathrm{e}}^{\mathrm{cota}}\end{array}$