The largest value of non-negative integer a for which lim x 1 { - a x + sin ( x - 1 ) + a x + sin ( x - 1 ) - 1 } 1 - x 1 - x = 1 4 is [2014]

Question

The largest value of non-negative integer $a$ for which $\lim_{x \to 1} {\frac{- a x + \sin (x - 1) + a}{x + \sin (x - 1) - 1}}^{\frac{1 - x}{1 - \sqrt{x}}} = \frac{1}{4}$ is [2014]

Smart Achievers · Accepted Answer

(2)

$\lim_{x \to 1} {\frac{- a x + \sin (x - 1) + a}{x + \sin (x - 1) - 1}}^{\frac{1 - x}{1 - \sqrt{x}}} = \frac{1}{4}$

$\Rightarrow \lim_{x \to 1} {\frac{a (1 - x) + \sin (x - 1)}{(x - 1) + \sin (x - 1)}}^{1 + \sqrt{x}}$

$\Rightarrow \lim_{x \to 1} {\frac{- a + \frac{\sin (x - 1)}{x - 1}}{1 + \frac{\sin (x - 1)}{x - 1}}}^{1 + \sqrt{x}}$

$\Rightarrow {(\frac{- a + 1}{2})}^{2} = \frac{1}{4}$

$\Rightarrow a = 0 or 2$

$∴$ $Largest value of a is 2 .$

Solution

Q.
The largest value of non-negative integer $a$ for which $\lim_{x \to 1} {\frac{- a x + \sin (x - 1) + a}{x + \sin (x - 1) - 1}}^{\frac{1 - x}{1 - \sqrt{x}}} = \frac{1}{4}$ is [2014]

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Solution

Q. The largest value of non-negative integer a for which limx→1{-ax+sin(x-1)+ax+sin(x-1)-1}1-x1-x=14 is [2014]

Ans. (2) limx→1{-ax+sin(x-1)+ax+sin(x-1)-1}1-x1-x=14 ⇒limx→1{a(1-x)+sin(x-1)(x-1)+sin(x-1)}1+x ⇒limx→1{-a+sin(x-1)x-11+sin(x-1)x-1}1+x ⇒(-a+12)2=14 ⇒a=0 or 2 ∴ Largest value of a is 2.

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Q.
The largest value of non-negative integer $a$ for which $\lim_{x \to 1} {\frac{- a x + \sin (x - 1) + a}{x + \sin (x - 1) - 1}}^{\frac{1 - x}{1 - \sqrt{x}}} = \frac{1}{4}$ is [2014]