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Find the limit

$\operatorname { lit } _ { x \rightarrow 1 } \left\{ \left( \frac { 1 + x } { 2 + x } \right) ^ { \left( \frac { 1 - \sqrt { x } } { 1 - x } \right) } \right\}$

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$\exp\lim_{x \to 1}\ln ((1+x)/(2+x))^{(1-\sqrt{x}/1-x)} $

$\exp \lim_{x\to 1 }((1-\sqrt{x})/(1-x))\ln ((1+x)/(2+x))$

 

$(1-\sqrt{x}/1-x)$ forms 0/0 forms when x=1 we can apply LHospital rule here So,

$\exp((-0.5)*(x\tfrac{-3}{2})/-1)*\ln (2/3)$

Putting the value of x=1

$\exp \ln (2/3)^{0.5}$

Answer will be$\sqrt{2/3}$

edited by

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Answer is 1. I also know the procedure. My question is why isn't this formula working?Am I doing something wrong?