The largest value of the non-negative integer a for which is

Engineering

Mathematics

One to The Power Infinity Form

Question

The largest value of the non-negative integer a for which $\underset{x \to 1}{Lim} {\frac{- ax + \sin (x - 1) + a}{x + \sin (x - 1) - 1}}^{\frac{1 - x}{1 - \sqrt{x}}} = \frac{1}{4}$ is

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a ∈ W

$\underset{x \to 1}{Lim} \frac{1 - x}{1 - \sqrt{x}} = \underset{x \to 1}{Lim} (1 + \sqrt{x}) = 2$

$∴ \underset{x \to 1}{Lim} {(\frac{- ax + \sin (x - 1) + a}{x + \sin (x - 1) - 1})}^{2} = \frac{1}{4}$

$\Rightarrow {(\frac{- a + 1}{2})}^{2} = 1 \Rightarrow$ a = 0, 2 (a = 2 rejected because base is negative at a = 2)

∴ a_largest = 0