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  4. displaystyle limx→ (π/2) ( cos x/ x - (π)2) equals:

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Q. $\displaystyle\lim_{x\to \frac{\pi}{2}} \frac{\cos \, x}{ x - \frac{\pi}{2}}$ equals:

Limits and Derivatives

Solution:

Let $ f(x ) = \displaystyle\lim_{x\to \frac{\pi}{2}} \frac{\cos \, x}{ x - \frac{\pi}{2}}$
Put $x - \frac{\pi}{2} = h$
When $x \to \pi /2 , h \to 0$
$ \therefore \:\:\: \displaystyle\lim_{h \to 0} \frac{\cos \left(\frac{\pi}{2} + h \right)}{h} = - \displaystyle\lim_{h \to 0} \frac{ \sin \, h }{h} = -1$