The value of limx→1 log⁡xsin⁡πx, is

# The value of $\underset{x\to 1}{lim} \frac{\mathrm{log}x}{\mathrm{sin}\pi x},$ is

1. A

$\frac{1}{\pi }$

2. B

$-\pi$

3. C

$\pi$

4. D

$-\frac{1}{\pi }$

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### Solution:

We have,

$\begin{array}{l}\underset{x\to 1}{lim} \frac{\mathrm{log}x}{\mathrm{sin}\pi x}=\underset{x\to 1}{lim} \frac{\mathrm{log}\left[1+\left(x-1\right)\right\}}{\mathrm{sin}\left(\pi -\pi x\right)}\\ =\underset{x\to 1}{lim} \frac{\mathrm{log}\left\{1+\left(x-1\right)\right\}}{\mathrm{sin}\pi \left(1-x\right)}\\ =\underset{x\to 1}{lim} \frac{\mathrm{log}\left\{1+\left(x-1\right)\right\}}{x-1}×\frac{x-1}{\mathrm{sin}\pi \left(1-x\right)}\\ =-\frac{1}{\pi }\underset{x\to 1}{lim} \frac{\mathrm{log}\left\{1+\left(x-1\right)\right\}}{x-1}×-\frac{\pi \left(1-x\right)}{\mathrm{sin}\pi \left(1-x\right)}=-\frac{1}{\pi }\end{array}$

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