Finding Limits in Simple Cases | 简单情形下的极限求解

📚 Finding Limits in Simple Cases | 简单情形下的极限求解

A limit is one of the most important ideas in calculus. It describes what happens to a function as the input approaches a particular value, without necessarily reaching it. In this article, we focus on simple cases where limits can be found by substitution, basic algebra, or by looking at the behaviour of a function as the input grows very large.

极限是微积分中最重要的概念之一。它描述的是当自变量趋近某个值（但不一定等于该值）时，函数的变化趋势。在本文中，我们聚焦于简单情形：可以通过代入法、基础代数化简或考察自变量越来越大时函数的行为来求出极限。

1. The Intuitive Idea of a Limit | 极限的直观思想

Suppose we have a function f(x). As x gets closer and closer to a value a, the outputs f(x) may get closer and closer to some number L. That number L is called the limit of f(x) as x approaches a. The function does not even need to be defined at x = a; we only care about the values near a.

假设我们有一个函数 f(x)。当 x 越来越接近某个数值 a 时，输出 f(x) 可能会越来越接近某个数 L。这个数 L 就称为 f(x) 当 x 趋于 a 时的极限。函数在 x = a 处甚至不需要有定义，我们只关心 a 附近的取值。

lim (x→a) f(x) = L

2. Notation and Basic Rules | 记号与基本法则

The expression lim (x→a) f(x) = L is read as ‘the limit of f(x) as x tends to a is L’. Limits obey simple arithmetic rules: the limit of a sum is the sum of the limits; the limit of a product is the product of the limits; the limit of a quotient is the quotient of the limits (provided the denominator limit is not zero).

表达式 lim (x→a) f(x) = L 读作“当 x 趋于 a 时 f(x) 的极限是 L”。极限遵循简单的算术法则：和的极限等于极限的和；积的极限等于极限的积；商的极限等于极限的商（前提是分母的极限不为零）。

lim (x→a) [f(x) + g(x)] = lim (x→a) f(x) + lim (x→a) g(x)

lim (x→a) [f(x) + g(x)] = lim (x→a) f(x) + lim (x→a) g(x)
lim (x→a) [c f(x)] = c lim (x→a) f(x), where c is a constant.

lim (x→a) [c f(x)] = c lim (x→a) f(x)，其中 c 为常数。
lim (x→a) [f(x) g(x)] = (lim (x→a) f(x)) (lim (x→a) g(x)).

lim (x→a) [f(x) g(x)] = (lim (x→a) f(x)) (lim (x→a) g(x))。

3. Direct Substitution for Polynomials | 多项式中的直接代入

Polynomials are continuous everywhere, so the limit as x→a is simply the value of the polynomial at a. We can find the limit by substituting x = a directly.

多项式处处连续，因此当 x→a 时，极限就等于多项式在 a 处的值。我们只要把 x = a 直接代入即可。

Example: lim (x→2) (x³ – 4x + 5) = 2³ – 4×2 + 5 = 8 – 8 + 5 = 5.

例：lim (x→2) (x³ – 4x + 5) = 2³ – 4×2 + 5 = 8 – 8 + 5 = 5。

This works for any polynomial. Other continuous functions, such as eˣ, sin x and cos x, also allow direct substitution at every point in their domains.

这对任何多项式都成立。其他连续函数，例如 eˣ、sin x 和 cos x，在定义域内每一点也可以直接代入。

4. Rational Functions and the 0/0 Form | 有理函数与 0/0 型

When we substitute x = a into a rational function and get 0/0, the limit is not obvious. In simple cases we factorise the numerator and denominator, cancel the common factor, and then substitute again.

当我们将 x = a 代入有理函数得到 0/0 时，极限并不明显。在简单情形下，我们将分子分母因式分解，约去公因子，然后再次代入。

Example: lim (x→3) (x² – 9)/(x – 3). Direct substitution gives 0/0. Factorise: (x² – 9) = (x – 3)(x + 3). After cancelling (x – 3), we have lim (x→3) (x + 3) = 6.

例：lim (x→3) (x² – 9)/(x – 3)。直接代入得到 0/0。因式分解：x² – 9 = (x – 3)(x + 3)。约去 (x – 3) 后，得到 lim (x→3) (x + 3) = 6。

The original function is not defined at x = 3, but the limit exists. This is called a removable discontinuity or a ‘hole’ in the graph.

原函数在 x = 3 处没有定义，但极限存在。这称为可去间断点，即图像上的“空洞”。

5. Using Common Factors | 利用公因子化简

Sometimes the numerator or denominator contains a common factor that is not easy to see, such as a difference of cubes or a quadratic that can be factored. A good first step is to simplify the expression algebraically before taking the limit.

有时分子或分母中含有不太容易看出的公因子，例如立方差或可因式分解的二次式。一个有效的做法是：在求极限之前先对表达式进行代数化简。

Example: lim (x→1) (x³ – 1)/(x – 1). Since x³ – 1 = (x – 1)(x² + x + 1), the limit becomes lim (x→1) (x² + x + 1) = 3.

例：lim (x→1) (x³ – 1)/(x – 1)。因为 x³ – 1 = (x – 1)(x² + x + 1)，所以极限变为 lim (x→1) (x² + x + 1) = 3。

6. Limits at Infinity | 无穷远处的极限

We sometimes need to know what happens to f(x) as x tends to +∞ or -∞. For polynomials, the term with the highest power dominates. For example, as x→∞, 4x³ + 2x – 1 behaves like 4x³, because the lower order terms become insignificant.

有时我们需要知道当 x 趋于 +∞ 或 -∞ 时 f(x) 的行为。对于多项式，最高次幂的项起主导作用。例如，当 x→∞ 时，4x³ + 2x – 1 的行为类似于 4x³，因为低阶项变得微不足道。

For rational functions, a reliable method is to divide both the numerator and denominator by the highest power of x that appears in the denominator.

对于有理函数，一种可靠的方法是：用分母中出现的 x 的最高次幂同时去除分子和分母。

Example: lim (x→∞) (3x² + 2)/(2x² – x). Divide every term by x²: (3 + 2/x²)/(2 – 1/x). As x→∞, 2/x² and 1/x approach 0, so the limit is 3/2.

例：lim (x→∞) (3x² + 2)/(2x² – x)。将每一项除以 x²：(3 + 2/x²)/(2 – 1/x)。当 x→∞ 时，2/x² 和 1/x 趋于 0，因此极限为 3/2。

7. Rules for Rational Functions at Infinity | 有理函数在无穷远处的一般规律

For a rational function where the numerator has degree m and the denominator has degree n, the limit as x→∞ can be classified as follows:

对于有理函数，若分子次数为 m，分母次数为 n，那么当 x→∞ 时的极限可按以下规则分类：

Situation 情形	Limit 极限
m > n	+∞ or -∞ (diverges)
m = n	ratio of leading coefficients
m < n	0

The same rules apply for x→-∞, though the sign of the leading power may change the direction.

同样的规则也适用于 x→-∞，只是最高次幂的符号可能改变发散的方向。

8. Infinite Limits and Vertical Asymptotes | 无穷极限与竖直渐近线

If f(x) grows without bound as x approaches a certain value a, we say the limit is infinite. A common simple case is f(x) = 1/(x – 2). As x→2⁺ (from the right), 1/(x – 2) → +∞; as x→2⁻ (from the left), 1/(x – 2) → -∞.

如果当 x 趋于某个值 a 时，f(x) 无限增大，我们就称极限为无穷。一个简单的常见例子是 f(x) = 1/(x – 2)。当 x→2⁺（从右侧接近）时，1/(x – 2) → +∞；当 x→2⁻（从左侧接近）时，1/(x – 2) → -∞。

To determine the sign, pick a value just above or just below a and evaluate the denominator. For example, with x = 2.1, the denominator is positive (0.1), so the function is positive, giving +∞.

要判断符号，可以在 a 的左侧或右侧取一个接近的值，代入分母判断正负。例如取 x = 2.1，分母为正（0.1），所以函数为正，极限为 +∞。

9. The Special Trigonometric Limit | 重要三角极限

One of the most important limits in A-Level mathematics is lim (θ→0) sin θ / θ = 1. It is used to differentiate sine and to find other trigonometric limits. The proof is based on a geometric argument using the unit circle, but in simple cases we can apply the result directly.

A-Level 数学中最重要的极限之一是 lim (θ→0) sin θ / θ = 1。它用于对正弦函数求导以及求其他三角极限。其证明基于单位圆的几何论证，但在简单情形下我们可以直接应用该结论。

For example, lim (θ→0) (sin 3θ)/θ = 3·lim (θ→0) (sin 3θ)/(3θ) = 3×1 = 3.

例如，lim (θ→0) (sin 3θ)/θ = 3·lim (θ→0) (sin 3θ)/(3θ) = 3×1 = 3。

Another useful result is lim (θ→0) (cos θ – 1)/θ = 0, which follows from the half-angle identity.

另一个有用结论是 lim (θ→0) (cos θ – 1)/θ = 0，它可由半角公式得到。

10. Connection to Differentiation from First Principles | 与“从第一原理”求导的联系

The derivative of a function f at a point a is defined by a limit:

函数 f 在点 a 处的导数通过如下极限定义：

f′(a) = lim (h→0) [f(a + h) – f(a)] / h

For simple polynomial functions, this limit can be found by expanding and simplifying. For example, if f(x) = x², then f′(3) = lim (h→0) [(3 + h)² – 9]/h = lim (h→0) (6h + h²)/h = lim (h→0) (6 + h) = 6.

对于简单多项式，该极限可以通过展开并化简来求。例如，若 f(x) = x²，则 f′(3) = lim (h→0) [(3 + h)² – 9]/h = lim (h→0) (6h + h²)/h = lim (h→0) (6 + h) = 6。

This limit process is the foundation of calculus: every derivative is a limit of a gradient of a secant line.

这个极限过程是微积分的基础：每个导数都是割线斜率在 h→0 时的极限。

11. Common Mistakes and Tips | 常见错误与技巧

Do not confuse the limit with the value of the function. A function may have a limit at a point where it is undefined.

不要将极限与函数值混淆。函数在未定义的点处也可能有极限。
If direct substitution gives 0/0, try to factorise and cancel before substituting again.

如果直接代入得到 0/0，先因式分解并约分，然后再代入。
For limits at infinity of rational functions, compare the highest powers in the numerator and denominator.

对于有理函数在无穷远处的极限，比较分子和分母的最高次幂。
For piecewise functions, check one-sided limits before concluding that a limit exists.

对于分段函数，要先检查单侧极限，再判断极限是否存在。

12. Summary | 小结

In simple cases, limits can be evaluated using direct substitution, algebraic simplification, or dominant-term analysis at infinity. Special limits like sin θ / θ and the definition of the derivative help connect these ideas to the wider calculus toolkit. Mastery of these basic techniques will prepare you for harder problems involving differentiation and integration.

简单情形下，极限可以通过直接代入、代数化简或在无穷远处对主导项的分析来求出。像 sin θ / θ 以及导数定义这样的特殊极限，将这些问题与更广的微积分工具联系起来。掌握这些基本技巧，将为解决涉及微分与积分的更困难问题做好准备。

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Finding Limits in Simple Cases | 简单情形下的极限求解