Using Series Expansions to Find Limits | 用级数展开求极限

📚 Using Series Expansions to Find Limits | 用级数展开求极限

When a limit produces an indeterminate form such as 0/0, series expansions often reveal the finite ratio hiding inside the competing small terms. In AQA A-Level Mathematics, Maclaurin expansions give a systematic way to replace functions by polynomials near x = 0, making limits quick and exact.

当极限出现 0/0 等未定式时，级数展开通常能揭示隐藏在相互抵消的小量中的有限比值。在 AQA A-Level 数学中，麦克劳林展开提供了一种把函数在 x = 0 附近替换为多项式的方法，使极限计算快速且精确。

1. The Core Idea: Replace with Local Polynomials | 核心思想：用局部多项式替换

Near x = 0, a smooth function can be written as a constant plus linear, quadratic and higher powers of x. When a limit gives 0/0, the constant terms vanish, so the leading non-zero power of x controls the value.

在 x = 0 附近，光滑函数可以写成常数项加 x 的一次、二次及更高次幂。当极限为 0/0 时，常数项相互抵消，因此第一个非零的 x 幂决定极限值。

Series expansion turns a limit of transcendental functions into a limit of polynomials, where cancellation is visible term by term.

级数展开把超越函数的极限转化为多项式的极限，从而使每一项的抵消都清晰可见。

f(x) = a₀ + a₁x + a₂x² + a₃x³ + ⋯ near x = 0

2. Standard Maclaurin Expansions | 标准麦克劳林展开

You should know these expansions from the AQA formulae booklet, especially their first three or four terms. They are the main tool for limits as x → 0.

你应当熟记 AQA 公式手册中的下列展开式，尤其是前三到四项。它们是处理 x → 0 型极限的主要工具。

Use them only around x = 0. If the limit is at another point, first shift the variable to bring it to zero.

这些展开式只适用于 x = 0 附近。如果极限在别的点，应先作变量平移，使其变为零。