• $\begin{array}{rl}& \underset{n\to \mathrm{\infty }}{lim}\sqrt[n]{a}=1(a>0)\\ & \underset{n\to \mathrm{\infty }}{lim}\sqrt[n]{n}=1\end{array}\}\sqrt[n]{n}=y_n+1$
  • [[2023-09-22 例题1]]
• $\underset{n\to 0}{lim}n\sin {\displaystyle \frac{1}{n}}=0$ 无穷小乘有界量
• $\underset{n\to \mathrm{\infty }}{lim}{\displaystyle \frac{a_0{n}^k+a_1{n}^{k-1}+\cdots +a_{k-1}n+a_k}{b_0{n}^m+b_1{n}^{m-1}+\cdots +b_{m-1}n+b_m}}=\{\begin{array}{ll}0,& k<m\\ {\displaystyle \frac{a_0}{b_0}},& k=m\\ \mathrm{\infty },& k>m\end{array}$
  • 提出来 $=\underset{n\to \mathrm{\infty }}{lim}{\displaystyle \frac{n^k(\cdots )}{n^m(\cdots )}}$
  • 倒置代换 $n\to \mathrm{\infty }⟹t={\displaystyle \frac{1}{n}}\to 0^{+}$
• $a\ge b\ge c>0,\underset{n\to \mathrm{\infty }}{lim}\sqrt[n]{a^n+b^n+c^n}=a$
  • 夹逼定理 $a=\sqrt[n]{a^n}\le \underset{n\to \mathrm{\infty }}{lim}\sqrt[n]{a^n+b^n+c^n}\le \sqrt[n]{3a^n}=a\sqrt[n]{3}=a$