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Differentiation of Inverse Trigonometric Functions

Solve lim x...

Question

Solve ${lim}_{x \to 0} \frac{\sqrt{1 + sin x} - \sqrt{1 - sin x}}{x}$

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Solution

${lim}_{x \to 0} \frac{\sqrt{1 + sin x} - \sqrt{1 - sin x}}{x}$
This is $\frac{0}{0}$ form so by using L hospital rule
${lim}_{x \to 0} \frac{- \frac{c o s x}{\sqrt{1 + sin x}} - \frac{c o s x}{\sqrt{1 - sin x}}}{1} = - 2$

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Q. Evaluate the following limit:

lim x \to 0 \frac{\sqrt{(1 + sin x)} - \sqrt{(1 - sin x)}}{x}

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