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  • Evaluate    <img alt="\mathrm{\lim _{x \rightarrow a}\left(\frac{\sin (x)}{\sin (a)}\right)^{\frac{1}{x-a}} }" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim

Evaluate    \mathrm{\lim _{x \rightarrow a}\left(\frac{\sin (x)}{\sin (a)}\right)^{\frac{1}{x-a}} }

Option: 1

\mathrm{e^{\frac{\cos a}{\sin a}}}


Option: 2

0


Option: 3

1


Option: 4

None


Answers (1)

best_answer

You can manipulate the expression, so that
\mathrm{ \lim _{x \rightarrow a}\left(1+\frac{1}{\frac{\sin a}{\sin x-\sin a}}\right)^{\frac{\sin a}{\sin x-\sin a} \frac{\sin x-\sin a}{\sin a} \frac{1}{x-a}} }


which is  \mathrm{\lim _{x \rightarrow a} e^{\frac{\sin x-\sin a}{x-a}} \frac{1}{\sin a}=e^{\frac{\cos a}{\sin a}} }

This is how the exponent becomes cot a
Edit: Maybe I skipped a few too many steps You can achieve the upper one expression by adding and subtracting one. So
\mathrm{ 1+\frac{\sin x}{\sin a}-1=1+\frac{\sin x-\sin a}{\sin a}=1+\frac{1}{\frac{\sin a}{\sin x-\sin a}} }

The exponent is
\mathrm{ 1 \cdot \frac{1}{x-a}=\frac{\sin a}{\sin x-\sin a} \frac{\sin x-\sin a}{\sin a} \frac{1}{x-a} }

Posted by

shivangi.shekhar

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