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Question Number 15181 by Joel577 last updated on 08/Jun/17

$$f\left(x\right)=\frac{(px+q).\sin 2x}{ax+b}$$$$\underset{x\to 0}{\lim }f\left(x\right)=2\mathrm{and}\underset{x\to \mathrm{\infty }}{\lim }f\left(x\right)=0$$$$\mathrm{Find}a,b,p,q$$

Answered by ajfour last updated on 08/Jun/17

$$f\left(h\right)=\frac{(ph+q)\left(2h\right)}{ah+b}\to 2$$$$\Rightarrow {ph}^2+qh\to ah+b$$$$\Rightarrow b=0anda=q$$$$f\left(\frac{1}{h}\right)=\frac{(p+qh)\sin \left(\frac{2}{h}\right)}{(a+bh)}\to 0$$$$\Rightarrow p+qh\to 0$$$$orp=0$$$$thusf\left(x\right)=\frac{\sin 2x}{x}.$$

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