Limits: Concept and Common Calculation Methods | 极限的概念与常用计算方法

📚 Limits: Concept and Common Calculation Methods | 极限的概念与常用计算方法

The concept of a limit is the foundation of calculus. In the IB Mathematics Analysis and Approaches course, limits are studied as a precursor to differentiation and integration. Understanding limits helps you interpret rates of change, asymptotes, and continuity. This article introduces the intuitive idea of a limit and presents the most common methods for evaluating limits, including substitution, factorisation, rationalisation, the squeeze theorem, limits at infinity, trigonometric limits, and L'Hopital's rule.

极限的概念是微积分的基础。在IB数学分析（Analysis and Approaches）课程中，极限是学习微分与积分前的预备知识。理解极限有助于解释变化率、渐近线和连续性。本文介绍极限的直观思想，并给出最常见的求极限方法，包括直接代入、因式分解、有理化、夹逼定理、无穷远处极限、三角极限及洛必达法则。

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1. The Intuitive Concept of a Limit | 极限的直观概念

Suppose a function f is defined near a point a. We ask: what value does f(x) get closer and closer to as x gets closer and closer to a? If the values f(x) approach a single number L, then L is called the limit of f at a. The notation is

limₓ→ₐ f(x) = L

and we say "the limit of f(x) as x approaches a equals L" not care what happens exactly at x = a; we only care what happens arbitrarily close to a.

设函数 f 在点 a 附近有定义。我们要问：当 x 无限接近 a 时，f(x) 无限接近哪个数？如果 f(x) 的值趋近于唯一的数 L，那么 L 就称为 f 在 a 处的极限，记作

limₓ→ₐ f(x) = L

并读作“当 x 趋近于 a 时，f(x) 的极限等于 L”。关键在于：我们并不关心 x = a 处本身发生了什么，只关心 a 附近无限接近时的趋势。

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2. Left-Hand and Right-Hand Limits | 左极限与右极限

For a limit to exist, the function must approach the same value from both directions. We write

limₓ→ₐ⁻ f(x) = L

for the left-hand limit, where x approaches a from values smaller than a, and

limₓ→ₐ⁺ f(x) = L

for the right-hand limit, where x approaches a from values larger than a. The two-sided limit exists only when the left-hand and right-hand limits are equal.

一个函数要在某点存在极限，必须从左右两个方向趋近于同一个值。记

limₓ→ₐ⁻ f(x) = L

为左极限，即 x 从小于 a 的一侧趋近 a；记

limₓ→ₐ⁺ f(x) = L

为右极限，即 x 从大于 a 的一侧趋近 a。只有当左右极限相等时，双侧极限才存在。

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3. When Does a Limit Not Exist? | 极限不存在的常见情形

A limit fails to exist when the left-hand and right-hand limits are different. For example, consider the piecewise function

f(x) = x + 1 if x < 2; f(x) = x² − 3 if x ≥ 2

x approaches 2 from the right, f(x) approaches 1. Since 3 ≠ 1, limₓ→₂ f(x) does not exist.

当左右极限不相等时，极限不存在。例如，考虑分段函数

f(x) = x + 1（若 x 2）；f(x) = x² − 3（若 x ≥ 2）

当 x 从左侧趋近 2 时，f(x) 趋近 3；当 x 从右侧趋近 2 时，f(x) 趋近 1。因为 3 ≠ 1，所以 limₓ→₂ f(x) 不存在。

x) as x tends to 0. The function oscillates infinitely often, so it settles on no single value.

另一种常见情形是无限震荡，例如当 x 趋近 0 时，f(x) = sin(1/x)。函数在 0 附近反复震荡，无法趋于一个确定的值。

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4. Direct Substitution | 直接代入法

For continuous functions such as polynomials, rational functions (where the denominator is not zero), exponential functions, and trigonometric functions, the limit can often be found by direct substitution. If f is continuous at a, then

limₓ→ₐ f(x) = f(a)

For example,

limₓ→₂ (3x² + x − 1) = 3(2)² + 2 − 1 = 12 + 1 = 13

s everywhere, the limit as x approaches 2 is exactly the value of the polynomial at x = 2.

对于连续函数，如多项式、分母不为零的有理函数、指数函数和三角函数，通常可以直接代入求极限。若 f 在 a 处连续，则

limₓ→ₐ f(x) = f(a)

例如，

limₓ→₂ (3x² + x − 1) = 3(2)² + 2 − 1 = 12 + 1 = 13

因为多项式处处连续，所以当 x 趋近 2 时的极限恰好等于该多项式在 x = 2 处的函数值。

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5. Factorisation and Cancellation | 因式分解与约分

When direct substitution gives an indeterminate form such as 0/0, try factorising the numerator and denominator. If a common factor exists, cancel it before evaluating the limit.

当直接代入出现 0/0 型不定式时，可以尝试对分子和分母进行因式分解。如果存在公因子，先约分再求极限。

Example: Evaluate

limₓ→₃ (x² − 9)/(x − 3)

Substituting x = 3 gives 0/0. Factorising gives

limₓ→₃ (x − 3)(x + 3)/(x − 3) = limₓ→₃ (x + 3) = 6

Once the common factor (x − 3) is cancelled, direct substitution works.

例：求极限

limₓ→₃ (x² − 9)/(x − 3)

代入 x = 3 得到 0/0。因式分解得

limₓ→₃ (x − 3)(x + 3)/(x − 3) = limₓ→₃ (x + 3) = 6

约去公因子 (x − 3) 后，直接代入即可。

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6. Rationalisation | 有理化方法

When the limit involves square roots, multiplying the numerator and denominator by the conjugate often eliminates the indeterminate form. For example, evaluate

limₓ→₀ (√x + 1 − 1)/x

Substituting x = 0 gives 0/0. Multiply numerator and denominator by the conjugate √x + 1 + 1:

limₓ→₀ [(√x + 1 − 1)(√x + 1 + 1)]/[x(√x + 1 + 1)]

Using the difference of squares, the numerator becomes x. Thus

limₓ→₀ x/[x(√x + 1 + 1)] = limₓ→₀ 1/(√x + 1 + 1) = 1/2

当极限表达式中含根号时，将分子分母同乘共轭式，常常可以消去不定式。例如，求

limₓ→₀ (√x + 1 − 1)/x

代入 x = 0 得到 0/0。将分子分母同乘共轭式 √x + 1 + 1：

limₓ→₀ [(√x + 1 − 1)(√x + 1 + 1)]/[x(√x + 1 + 1)]

利用平方差公式，分子变为 x，于是

limₓ→₀ x/[x(√x + 1 + 1)] = limₓ→₀ 1/(√x + 1 + 1) = 1/2

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7. The Squeeze Theorem | 夹逼定理

theorem states that if g(x) ≤ f(x) ≤ h(x) for all x near a (except possibly at a), and if

limₓ→ₐ g(x) = limₓ→ₐ h(x) = L

→ₐ f(x) = L. This method is especially useful for functions that oscillate or are bounded.

夹逼定理指出：若在 a 附近（除 a 点本身外）有 g(x) ≤ f(x) ≤ h(x)，并且

limₓ→ₐ g(x) = limₓ→ₐ h(x) = L

则 limₓ→ₐ f(x) = L。该方法特别适用于震荡或有界的函数。

e have −x² ≤ x² sin(1/x) ≤ x². Both −x² and x² tend to 0 as x → 0. By the squeeze theorem,

limₓ→₀ x² sin(1/x) = 0

例：求 limₓ→₀ x² sin(1/x)。因为 −1 ≤ sin(1/x) ≤ 1，所以 −x² ≤ x² sin(1/x) ≤ x²。当 x → 0 时，−x² 与 x² 都趋近于 0。由夹逼定理，

limₓ→₀ x² sin(1/x) = 0

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8. Limits at Infinity and Horizontal Asymptotes | 无穷远处的极限与水平渐近线

When x becomes very large,

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