21.   Evaluate the following limits \lim_{x \rightarrow 0} \left ( \csc x - \cot x \right )

Answers (1)

\lim_{x \rightarrow 0} \left ( \csc x - \cot x \right )

On putting the limit directly the function takes infinity by infinity form, So we simplify the function and then put the limit

\lim_{x \rightarrow 0} \left ( \csc x - \cot x \right )

=\lim_{x \rightarrow 0} \left (\frac{1}{sinx}-\frac{cosx}{sinx}\right )

=\lim_{x \rightarrow 0} \left (\frac{1-cosx}{sinx}\right )

=\lim_{x \rightarrow 0} \left (\frac{2sin^2(\frac{x}{2})}{sinx}\right )

=\lim_{x \rightarrow 0} \left (\frac{2sin^2(\frac{x}{2})}{(\frac{x}{2})^2}\right )\left ( \frac{(\frac{x}{2})^2}{sinx} \right )

=\lim_{x \rightarrow 0} \frac{2}{4}\left (\frac{sin^2(\frac{x}{2})}{(\frac{x}{2})^2}\right )\left ( \frac{(x)}{sinx} \right )\cdot x

=\frac{2}{4}\times (1)^2\times0

=0   (Answer)

 

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