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Question Number 216077 by MATHEMATICSAM last updated on 27/Jan/25

$$\mathrm{Find}\mathrm{the}\mathrm{largest}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{non}\mathrm{negative}$$$$\mathrm{integer}p\mathrm{for}\mathrm{which}$$$$\underset{x\to 1}{\lim }{\left\{\frac{-px+\sin (x-1)+p}{x+\sin (x-1)-1}\right\}}^{\frac{1-x}{1-\sqrt{x}}}=\frac{1}{4}.$$

Answered by mahdipoor last updated on 27/Jan/25

$$ln\left(y\right)=\left(\frac{1-x}{1-\sqrt{x}}\right)ln\left(\frac{sin(x-1)+p(x-1)}{sin(x-1)+(x-1)}\right)$$$$\Rightarrow \underset{x\to 1}{\lim }\frac{1-x}{1-\sqrt{x}}=1+\sqrt{x}=2$$$$\Rightarrow \underset{x\to 1}{\lim }\frac{sin(x-1)+p(x-1)}{sin(x-1)+(x-1)}=\frac{0}{0}\Rightarrow hopital\Rightarrow$$$$\Rightarrow =\frac{cos(x-1)+p}{cos(x-1)+1}=\frac{p+1}{2}$$$$\underset{x\to 1}{\lim }ln\left(y\right)=ln\left(\frac{1}{4}\right)=2\times ln\left(\frac{p+1}{2}\right)$$$$\Rightarrow p=0$$

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