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Question
\[\lim_{x \to 0} \left( \cos x \right)^{1/\sin x}\]
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Solution
\[\lim_{x \to 0} \left( \cos x \right)^\frac{1}{\sin x} \]
\[ = \lim_{x \to 0} \left[ 1 + \cos x - 1 \right]^\frac{1}{\sin x} \]
\[\text{ Using the theoremgiven below }:\]
\[If \lim_{x \to a} f\left( x \right) = \lim_{x \to a} g\left( x \right) = 0 \text{ such that } \lim_{x \to a} \frac{f\left( x \right)}{g\left( x \right)} \text{ exists, then } \lim_{x \to a} \left[ 1 + f\left( x \right) \right]^\frac{1}{g\left( x \right)} = e^\lim_{x \to a} \frac{f\left( x \right)}{g\left( x \right)} . \]
\[Here: \]
\[ f\left( x \right) = \cos x - 1\]
\[ g\left( x \right) = \sin x\]
\[ \Rightarrow e^\lim_{x \to 0} \left( \frac{\cos x - 1}{\sin x} \right) \]
\[ = e^\lim_{x \to 0} \left( \frac{- 2 \sin^2 \frac{x}{2}}{2 \sin \frac{x}{2} \cos \frac{x}{2}} \right) \]
\[ = e^\lim_{x \to 0} \left( - \tan\frac{x}{2} \right) \]
\[ = e^0 \]
\[ = 1\]
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