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Question

$$\displaystyle \underset{x\rightarrow \infty }{lim}x\cos \left ( \frac{\pi }{4x} \right )\sin \left ( \frac{\pi }{4x} \right )= \frac{\pi }{b}.$$ Find b


Solution

Substitute  $$\displaystyle \left ( \pi /4x \right )= y $$
$$\therefore y\rightarrow 0$$ when $$\displaystyle x\rightarrow \infty $$
Limit $$= \displaystyle \underset{y\rightarrow 0}{lim}\dfrac{\pi }{4y}\cos y\sin y= \underset{y\rightarrow 0}{lim}\dfrac{\pi }{4}.\cos y.\dfrac{\sin y}{y}$$
$$\displaystyle = \dfrac{\pi }{4}.1.= \dfrac{\pi }{4}.$$
$$\Rightarrow b=4$$

Mathematics

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