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The largest value of the non-negative i...

The largest value of the non-negative integer a for which ` lim_( x to 1) {(-ax+ sin (x-1) +a)/(x+sin (x-1)-1)}^((1-x)/(1-sqrtx))=1/4 ` is

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The correct Answer is:
`a = 2`
PLAN ` underset( x to 0) lim (sin x) / x = 1`
Given, ` underset( x to 1 ) lim {(sin (x-1) +a(1-x))/((x-1)+sin (x-1))}^(((1+sqrtx)(1-sqrtx))/(1-sqrtx)) = 1/4`
` underset( x to 1) lim {((sin (x-1))/((x-1))-a)/(1+(sin(x-1))/((x-1)))}^(1+sqrtx) = 1/4`
`rArr" " ((1-a)/2)^(2) = 1/4 rArr (a-1)^(2) = 1`
` rArr" " a = 2 or 0` ltbr. Hence, the maximum value of a is 2.
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