Question

Покажите как решать эту задачу

Answer 1

$lim_{n to infty} sqrt{n}cdot ( sqrt{n+2}- sqrt{n+3})= \ \ = lim_{n to infty} sqrt{n}cdot frac{( sqrt{n+2}- sqrt{n+3})( sqrt{n+2} +sqrt{n+3})}{ sqrt{n+2} +sqrt{n+3}}= \ \ = lim_{n to infty} sqrt{n}cdot frac{( sqrt{n+2})^2- (sqrt{n+3})^2}{ sqrt{n+2} +sqrt{n+3}}= \ \ = lim_{n to infty} sqrt{n}cdot frac{n+2- (n+3)}{ sqrt{n+2} +sqrt{n+3}}= \ \ = lim_{n to infty} sqrt{n}cdot frac{n+2- n-3}{ sqrt{n+2} +sqrt{n+3}}= \ \ =$

$= lim_{n to infty} frac{- sqrt{n} }{ sqrt{n+2} +sqrt{n+3}}=$

делим и числитель и знаменатель на √n

$lim_{n to infty} frac{ frac{- sqrt{n}}{ sqrt{n}} }{ frac{sqrt{n+2}}{ sqrt{n}} + frac{sqrt{n+3}}{ sqrt{n}} }= \ \ =$

$lim_{n to infty} frac{- 1}{ sqrt{ frac{n+2}{n} } + sqrt{ frac{n+3}{n} } }= \ \ =lim_{n to infty} frac{- 1}{ sqrt{1 +frac{2}{n} } + sqrt{ 1+frac{3}{n} } }= - frac{1}{2}$