The largest value of the non-negative integer a for which limα→1 −ax+sin⁡(x−1)+ax+sin⁡(x−1)−11−x1−x=14, is

# The largest value of the non-negative integer a for which $\underset{\alpha \to 1}{lim} {\left\{\frac{-ax+\mathrm{sin}\left(x-1\right)+a}{x+\mathrm{sin}\left(x-1\right)-1}\right\}}^{\frac{1-x}{1-\sqrt{x}}}=\frac{1}{4},$ is

1. A

2

2. B

3

3. C

4

4. D

5

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### Solution:

We have.

$\underset{\alpha \to 1}{lim} {\left\{\frac{-ax+\mathrm{sin}\left(x-1\right)+a}{x+\mathrm{sin}\left(x-1\right)-1}\right\}}^{\frac{1-x}{1-\sqrt{x}}}=\frac{1}{4}$

$⇒\underset{\alpha \to 1}{lim} {\left\{\frac{\frac{\mathrm{sin}\left(x-1\right)}{x-1}-a}{1+\frac{\mathrm{sin}\left(x-1\right)}{x-1}}\right\}}^{1+\sqrt{x}}=\frac{1}{4}$

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