📚 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。
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”. The crucial point is that we do 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。
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。
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
As x approaches 2 from the left, f(x) approaches 3. As 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。
Another common situation is unbounded oscillation, such as f(x) = sin(1/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。
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
Because a polynomial is continuous 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。
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。
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
7. The Squeeze Theorem | 夹逼定理
The squeeze 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
then limₓ→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。
Example: Evaluate limₓ→₀ x² sin(1/x). Since −1 ≤ sin(1/x) ≤ 1, we 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
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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