$\lim_{n\rightarrow\infty} n-\frac{\sqrt{n}}{\sqrt{n+2}-\sqrt{n+1}}

A cura di: Administrator

Il limite si presenta in forma indeterminata.

Risulta, razionalizzando il denominatore

$lim_{nrightarrowinfty} n-frac{sqrt{n}}{sqrt{n+2}-sqrt{n+1}}=$$lim_{nrightarrowinfty} n-frac{sqrt{n}cdot (sqrt{n+2}+sqrt{n+1})}{1}=$$lim_{nrightarrowinfty} n-bigg[sqrt{n}cdotsqrt{n}cdotbigg(sqrt{1+frac{2}{n}}+sqrt{1+frac{1}{n}}bigg)bigg]=$$lim_{nrightarrowinfty} ncdotBigg[1-bigg(sqrt{1+frac{2}{n}}+sqrt{1+frac{1}{n}}bigg)Bigg]= +inftycdot (-1)=-infty$