Question

The value of $\underset{x\to \frac{\pi }{2}}{lim}\frac{\mathrm{sinx}-{\left(\mathrm{sinx}\right)}^{\mathrm{sinx}}}{1-\mathrm{sinx}+\ell nsinx}$ is

• A. 1
• B. 2
• C. 3
• D. $\frac{\pi }{2}$

Answer 1

Answer: (B)

2

We have,

$\underset{x\to \pi /2}{lim}\frac{\sin x-{\left(\sin x\right)}^{\sin x}}{1-\sin x+\ell n\left(\sin x\right)}$

$\sin x=t$

$\begin{array}{c}x\to \frac{\pi }{2}\sin x\to 1t\to 1\\ {lim}_{t\to 1}\frac{t-t^t}{-t+\ell n\left(t\right)}\\ {lim}_{t\to 1}\frac{1-t^t\left[1-\log t\right]}{-1+\frac{1}{t}}\end{array}$

$\underset{t\to 1}{lim}\frac{t-t^{th}\left[1+\log t\right]}{1-t}$

$\underset{t\to 1}{lim}\frac{1-t^{th}\left[\frac{th}{t}+\log t\right]+\left[t^{th}\times \frac{1}{t}+\log t^{th}\left(\frac{th}{t}+\log t\right)\right]}{-1}$

$\frac{1-2-1}{-1}=2$

Hence,

Option $B$ is correct answer.