Question

\(\lim _{x \rightarrow 1}\left(\frac{1+x}{2+x}\right)^{\frac{(1-\sqrt{x})}{(1-x)}}\)
(A) is 1
(B) does not exist
(C) is \(\sqrt{\frac{2}{3}}\)
(D) is \(/ \mathrm{n} 2\)

Answer 1

Ans: (C)
\(: \lim _{x \rightarrow 1}\left(\frac{1+x}{2+x}\right)^{\frac{1-\sqrt{x}}{1-x}}=\lim _{x \rightarrow 1}\left(\frac{1+x}{2+x}\right)^{\frac{1}{1+\sqrt{x}}}=\left(\frac{1+1}{2+1}\right)^{\frac{1}{1+1}}=\left(\frac{2}{3}\right)^{\frac{1}{2}}\)