Question

Lim∞ √((n+2)(n+1) )-√((n+3)n)

Answer 1

Домножим и поделим на сопряженное к √((n+2)(n+1) )-√((n+3)n)

$\displaystyle \lim_{n \to \infty}\dfrac{\left(\sqrt{(n+2)(n+1)}-\sqrt{n(n+3)}\right)\left(\sqrt{(n+2)(n+1)}+\sqrt{n(n+3)}\right)}{\sqrt{(n+2)(n+1)}+\sqrt{n(n+3)}}=\\ \\ \\ =\lim_{n \to \infty}\dfrac{(n+2)(n+1)+n(n+3)}{\sqrt{n^2+3n+2}+\sqrt{n^2+3n}}=\lim_{n \to \infty}\dfrac{n^2+3n+2-n^2-3n}{\sqrt{n^2+3n+2}+\sqrt{n^2+3n}}=\\ \\ \\ =\lim_{n \to \infty}\dfrac{2}{\sqrt{n^2+3n+2}+\sqrt{n^2+3n}}=\lim_{n \to \infty}\dfrac{2/n}{\sqrt{1+3/n+2/n^2}+\sqrt{1+3/n}}=0$