Question

пределы первый столбец 3, второй столбец 2,3​

Answer 1

$1)\ \lim\limits _{x \to \infty}\Big(\dfrac{4x+5}{4x-1}\Big)^{x+3}=\lim\limits _{x \to \infty}\Big(\Big(1+\dfrac{6}{4x-1}\Big)^{\frac{4x-1}{6}}\Big)^{\frac{6(x+3)}{4x-1}}=e^{\frac{6}{4}}=e^{\frac{3}{2}}$

$2)\ \ \lim_{n \to \infty}\dfrac{3n^{4/3}+\sqrt[3]{n^2-1}}{(n+\sqrt{n})^2}=\Big[\ \dfrac{:n^2}{:n^2}\ \Big]=\lim\limits _{n \to \infty}\dfrac{\frac{3}{n^{2/3}}+\sqrt[3]{\frac{1}{n^4}-\frac{1}{n^6}}}{1+\frac{2}{\sqrt{n}}+\frac{1}{n}}=\dfrac{0}{1}=0$

$3)\ \ \lim\limits _{x \to 0}\dfrac{ln(1+sinx)}{e^{x^2}-1}=\lim\limits _{x \to 0}\dfrac{sinx}{x^2}=\lim\limits _{x \to 0}\Big(\dfrac{sinx}{x}\cdot \dfrac{1}{x}\Big)=\lim\limits _{x \to 0}\Big(1\cdot \dfrac{1}{x}\Big)=\infty$