求极限方法总结

常用等价无穷小代换

当 \(t\to0\)

\begin{align}
\tan t &\sim t \\
\sin t & \sim t \\
1-\cos ^2 t &\sim \frac{x^2}{2} \\
(1+t)^k-1 &\sim kt \\
(e^t – 1) & \sim t \\
\ln(1+t) & \sim t
\end{align}

计算和幂函数的等价无穷小方法:寻找 \(x^k\) , 使

\begin{align}
\lim_{x \to 0}\frac{f(x)}{x^k}=c
\end{align}

则 \(f(x)\) 等价 \(cx^k\)

积分和变换

\begin{align}
\frac{1}{n}\sum_{i=0}^{n}f(\frac{i}{n}) &= \int_0^1f(x)dx \\
\frac{b-a}{n}\sum_{i=0}^{i=n}f(a+(b-a)\frac{i}{n}) & = \int_b^af(x)dx
\end{align}

Leave a Comment