Question

помогите пожалуйста ​

Answer 1

$1)\ \ \lim\limits _{n \to \infty}(\sqrt{n+1}-\sqrt{n})=\lim\limits _{n \to \infty}\dfrac{(n+1)-n}{\sqrt{n+1}+\sqrt{n}}= \lim\limits _{n \to \infty}\dfrac{1}{\sqrt{n+1}+\sqrt{n}}=0$

$2)\ \ \lim\limits_{x \to \infty}\dfrac{x^2-6x}{\sqrt{9x^4+2x-6}}=\Big[\ \dfrac{:x^2}{:x^2}\ \Big]=\lim\limits_{x \to \infty}\dfrac{1-\frac{6}{x}}{\sqrt{9+\frac{2}{x^3}-\frac{6}{x^4}}}=\dfrac{1}{\sqrt9}=\dfrac{1}{3}$

$3)\ \ \lim\limits _{n \to \infty}\dfrac{\sqrt{n+2}-\sqrt{n-3}}{\sqrt{2n+5}-\sqrt{3n-6}}= \lim\limits_{n \to \infty}\dfrac{\sqrt{1+\frac{2}{n}}-\sqrt{1-\frac{3}{n}}}{\sqrt{2+\frac{5}{n}}-\sqrt{3-\frac{6}{n}}}=\dfrac{1-1}{\sqrt2-\sqrt3}=0$