What is the limit where $lim_(x → 1) (sqrt(3+x)-sqrt(5-x))/(x^2-1)$ ?

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$(sqrt(3+x)-sqrt(5-x))/(x^2-1)$

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Answer 1 · 2018-04-05T14:28:29

$1/4$ .

$"The Reqd. Lim."=lim_(x to 1){sqrt(3+x)-sqrt(5-x)}/(x^2-1)$ ,

$=lim{sqrt(3+x)-sqrt(5-x)}/(x^2-1)xx{sqrt(3+x)+sqrt(5-x)}/{sqrt(3+x)+sqrt(5-x)}$ ,

$=lim{(3+x)-(5-x)}/{(x^2-1)(sqrt(3+x)+sqrt(5-x))}$ ,

$=lim{2cancel((x-1))}/{cancel((x-1))(x+1)(sqrt(3+x)+sqrt(5-x))}$ ,

$=lim_(x to 1)2/{(x+1)(sqrt(3+x)+sqrt(5-x))}$ ,

$=2/{(1+1)(sqrt(3+1)+sqrt(5-1))$ .

$rArr "The Reqd. Lim."=1/4$ .