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Q. $\displaystyle \lim_{x \to \frac{\pi }{2}}$ $\frac{1-cos\,x}{sin\,x}$ is

Limits and Derivatives

Solution:

We have, $\displaystyle \lim_{x \to \frac{\pi }{2}}$ $\frac{1-cos\,x}{sin\,x}$
$=\displaystyle \lim_{x \to \frac{\pi }{2}}$ $\frac{2\,sin^{2} \left(\frac{x}{2}\right)}{2\,sin \left(\frac{x}{2}\right)cos\left(\frac{x}{2}\right)}$
$=\displaystyle \lim_{x \to \frac{\pi }{2}}$$tan \left(\frac{x}{2}\right)$
$=tan \left(\frac{\pi}{4}\right)=1$