Finding Limits in Simple Cases | 简单情形下求极限

📚 Finding Limits in Simple Cases | 简单情形下求极限

In A-level calculus, a limit answers a simple question: where is a function heading? We write f(x) → L as x → a to mean that the output values of f(x) get arbitrarily close to the number L whenever x is close enough to a, even if the function is not actually defined at a.

在 A-level 微积分中，极限回答一个简单问题：函数正趋向何处？我们写 f(x) → L (当 x → a)，表示只要 x 足够接近 a，f(x) 的输出值就会任意接近数 L，即使函数在 a 点实际上没有定义。

For example, consider f(x) = (x² − 4)/(x − 2). This function is undefined at x = 2, but as x approaches 2 from either side, the output values approach 4. We therefore say the limit is 4.

例如，考虑 f(x) = (x² − 4)/(x − 2)。这个函数在 x = 2 处没有定义，但当 x 从两侧趋近 2 时，输出值都趋近 4。因此我们说极限为 4。

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1. What Is a Limit? | 什么是极限？

The limit of a function describes the value that f(x) approaches as x gets closer and closer to a particular number. The limit does not depend on the actual value of f at that point, because the point itself may be excluded from the domain.

函数的极限描述的是当 x 越来越接近某个特定数值时，f(x) 所趋近的值。极限不依赖于函数在该点的实际值，因为该点本身可能不在定义域内。

In AQA A-level Mathematics, we use an intuitive definition: if we can make f(x) as close as we like to L by taking x sufficiently close to a, then lim (x → a) f(x) = L. This informal idea is enough for evaluating simple limits algebraically.

在 AQA A-level 数学中，我们使用直观定义：如果让 x 足够接近 a，就能使 f(x) 任意接近 L，那么就记作 lim (x → a) f(x) = L。这一直观概念足以帮助我们通过代数方法计算简单极限。

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2. Notation and Key Terms | 记号与关键术语

The standard notation is lim (x → a) f(x) = L. This is read as ‘the limit of f(x) as x approaches a is L’. We also use the arrow form f(x) → L as x → a.

标准记号是 lim (x → a) f(x) = L，读作“当 x 趋近 a 时，f(x) 的极限为 L”。我们也使用箭头形式 f(x) → L (当 x → a)。

Key terms include ‘approaches’, ‘tends to’, ‘left-hand limit’ and ‘right-hand limit’. The left-hand limit is written lim (x → a⁻) f(x), meaning x approaches a from values less than a. The right-hand limit is lim (x → a⁺) f(x), meaning x approaches a from values greater than a.

关键术语包括“趋近”“趋向”“左极限”和“右极限”。左极限写作 lim (x → a⁻) f(x)，表示 x 从小于 a 的一侧趋近 a；右极限写作 lim (x → a⁺) f(x)，表示 x 从大于 a 的一侧趋近 a。

Understanding these phrases is essential because AQA exam questions often describe limits in words before asking for algebraic evaluation.

理解这些术语非常重要，因为 AQA 考试题经常会先用文字描述极限，再要求进行代数计算。

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3. Limits by Direct Substitution | 直接代入法求极限

If f is a polynomial, a rational function whose denominator is not zero at a, a root function defined at a, or a combination of these, then the simplest way to find the limit is to substitute x = a directly. For polynomial functions this always works because polynomials are continuous everywhere.

如果 f 是多项式、在 a 点分母不为零的有理函数、在 a 点有定义的根式函数，或这些函数的组合，那么求极限最简单的方法就是直接代入 x = a。对于多项式函数来说，这种方法总是有效，因为多项式处处连续。

Example: Find lim (x → 3) (2x² − 5x + 1). Substituting x = 3 gives 2(3)² − 5(3) + 1 = 18 − 15 + 1 = 4. Therefore the limit is 4.

例题：求 lim (x → 3) (2x² − 5x + 1)。代入 x = 3，得到 2(3)² − 5(3) + 1 = 18 − 15 + 1 = 4。因此极限为 4。

Direct substitution is also valid for trigonometric, exponential and logarithmic functions provided that the point is in the domain of the function. However, if substitution gives 0/0, ∞/∞ or another undefined expression, further algebra is needed.

直接代入法也适用于三角函数、指数函数和对数函数，只要该点在函数的定义域内。但是，如果代入后得到 0/0、∞/∞ 或其他未定义表达式，就需要进一步的代数处理。

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4. Indeterminate Form 0/0 and Factorisation | 0/0 未定式与因式分解

The most common A-level limit problem gives 0/0 after direct substitution. This does not mean the limit is automatically undefined; it means the function has a removable factor that must be cancelled. A limit of the form 0/0 is called an indeterminate form.

A-level 中最常见的极限问题是在直接代入后得到 0/0。这并不意味着极限一定不存在；它说明函数有一个可以约去的因子。形如 0/0 的极限称为未定式。

Example: Evaluate lim (x → 2) (x² − 4)/(x − 2). Direct substitution gives 0/0. Factor the numerator: x² − 4 = (x − 2)(x + 2), so the expression simplifies to x + 2, provided x ≠ 2. Then substitute x = 2 to obtain 2 + 2 = 4.

例题：求 lim (x → 2) (x² − 4)/(x − 2)。直接代入得到 0/0。将分子因式分解：x² − 4 = (x − 2)(x + 2)，因此表达式化简为 x + 2，只要 x ≠ 2。然后代入 x = 2，得到 2 + 2 = 4。

lim (x → 2) (x² − 4)/(x − 2) = 4

This works because the limit depends only on values of x close to 2 but not exactly equal to 2. Cancelling the factor (x − 2) is valid for every x ≠ 2 in the limiting process.

这种方法之所以有效，是因为极限只取决于 x 接近 2 但不等于 2 的值。在极限过程中，对于所有 x ≠ 2，约去因子 (x − 2) 都是合法的。

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5. Rationalising to Find Limits | 有理化求极限

When a limit involves a square root and direct substitution gives 0/0, multiplying the numerator and denominator by the conjugate can remove the problematic root. For example, if the numerator contains √x − 1, multiply by √x + 1.

当极限中含有平方根，并且直接代入得到 0/0 时，将分子和分母同时乘以共轭式可以消去引起问题的根号。例如，如果分子含有 √x − 1，就乘以 √x + 1。

Example: Evaluate lim (x → 1) (√x − 1)/(x − 1). Substitution gives 0/0. Multiply top and bottom by √x + 1:

例题：求 lim (x → 1) (√x − 1)/(x − 1)。代入得到 0/0。将分子和分母同时乘以 √x + 1：

(√x − 1)(√x + 1) / ((x − 1)(√x + 1)) = (x − 1)/((x − 1)(√x + 1)) = 1/(√x + 1)

This simplification is valid for x ≠ 1. Now substitute x = 1 to get 1/(√1 + 1) = 1/2.

这个化简对于 x ≠ 1 成立。现在代入 x = 1，得到 1/(√1 + 1) = 1/2。

Rationalising is especially useful in limits that define derivatives of square-root functions. In AQA papers, this technique appears regularly when the denominator is x − a and the numerator contains a square root.

有理化在定义平方根函数导数的极限中尤其有用。在 AQA 试卷中，当分母为 x − a 且分子含有平方根时，这种技巧经常出现。

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6. One-Sided Limits | 单侧极限

Some limits require checking the left and right behaviour separately. We write lim (x → a⁻) f(x) for the left-hand limit and lim (x → a⁺) f(x) for the right-hand limit. The two-sided limit exists only when these two one-sided limits are equal.

有些极限需要分别考察左侧和右侧的情况。我们用 lim (x → a⁻) f(x) 表示左极限，用 lim (x → a⁺) f(x) 表示右极限。只有当这两个单侧极限相等时，双侧极限才存在。

Example: For f(x) = |x|/x, the left-hand limit at x = 0 is lim (x → 0⁻) |x|/x = −1 because |x| = −x when x < 0. The right-hand limit is lim (x → 0⁺) |x|/x = 1 because |x| = x when x > 0. Since −1 ≠ 1, the two-sided limit lim (x → 0) |x|/x does not exist.

例题：对于 f(x) = |x|/x，在 x = 0 处的左极限为 lim (x → 0⁻) |x|/x = −1，因为当 x < 0 时 |x| = −x。右极限为 lim (x → 0⁺) |x|/x = 1，因为当 x > 0 时 |x| = x。由于 −1 ≠ 1，双侧极限 lim (x → 0) |x|/x 不存在。

One-sided limits are particularly important for piecewise functions and for rational functions with vertical asymptotes, where the function may diverge to +∞ on one side and −∞ on the other.

单侧极限对于分段函数以及具有竖直渐近线的有理函数尤为重要，因为这些函数可能在一侧趋向 +∞，在另一侧趋向 −∞。

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7. Limits at Infinity | 无穷远处的极限

When x → ∞ or x → −∞, we are interested in the long-run behaviour of a function. For rational functions, divide every term in the numerator and denominator by the highest power of x that appears in the denominator. Terms of the form c/xⁿ tend to 0 as x → ∞.

当 x → ∞ 或 x → −∞ 时，我们关注函数的长期变化趋势。对于有理函数，将分子和分母中的每一项都除以分母中出现的 x 的最高次幂。形如 c/xⁿ 的项当 x → ∞ 时趋于 0。

Example: Find lim (x → ∞) (3x² + 2x − 1)/(x² + 5). Divide top and bottom by x²:

例题：求 lim (x → ∞) (3x² + 2x − 1)/(x² + 5)。将分子和分母同时除以 x²：

(3 + 2/x − 1/x²)/(1 + 5/x²)

As x → ∞, 2/x → 0, 1/x² → 0 and 5/x² → 0, so the limit is 3/1 = 3.

当 x → ∞ 时，2/x → 0，1/x² → 0，5/x² → 0，因此极限为 3/1 = 3

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