Question

lim x стремится к 0 1-√1-x^2 знаменатель x^2

Answer 1

$lim_{x to0} frac{1- sqrt{1- x^{2} } }{ x^{2} } = frac{1- sqrt{1-0} }{0} = frac{0}{0}$
 неопределенность вида (0/0) 
(a-b)*(a+b)=a²-b²
(√a)²=a, a≥0

$= lim_{x to 0} frac{(1- sqrt{1- x^{2} } )*(1+ sqrt{1- x^{2} } )}{ x^{2} *(1+ sqrt{1+ x^{2} } )} = lim_{x to 0} frac{ 1^{2}-( sqrt{1- x^{2} } ) ^{2} }{ x^{2} *(1+ sqrt{1- x^{2} } )} =$
$= lim_{x to 0} frac{1-1+ x^{2} }{ x^{2} *(1+ sqrt{1- x^{2} } )} = lim_{x to 0} frac{ x^{2} }{ x^{2} *(1+ sqrt{1- x^{2} } )} = lim_{x to0} frac{1}{1+ sqrt{1- x^{2} } } =$
$= frac{1}{1+ sqrt{1- 0^{2} } } = frac{1}{2} =0,5$