f(x) = sin xsin a1x -a if x≠ a and f(x) = k if x = a and f(x) is continuous at x = a then k =

limx→af(x) =limx→a sin xsin a1x -a =elimx→a sin xsin a-11x -a =elimx→a sin x-sinax -a1sin a =elimx→a cos x-011sin a =ecota

f(x) = sin xsin a1x -a if x≠ a and f(x) = k if x = a and f(x) is continuous at x = a then k =

limx→af(x) =limx→a sin xsin a1x -a =elimx→a sin xsin a-11x -a =elimx→a sin x-sinax -a1sin a =elimx→a cos x-011sin a =ecota

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

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$e^{1 / a}$

$e^{\tan a}$

$e^{c o t a}$

$e^{a}$

answer is B.

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Detailed Solution

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