Limits at Infinity | 无穷远处的极限

📚 Limits at Infinity | 无穷远处的极限

In IB Mathematics, particularly in the Analysis and Approaches (AA) course, understanding the behaviour of functions as the input grows arbitrarily large (or negatively large) is essential. This concept, known as limits at infinity, helps us identify horizontal asymptotes, compare growth rates of different functions, and analyse long-term trends in mathematical models. Mastery of limits at infinity lays the foundation for calculus topics such as improper integrals and series convergence.

在IB数学中，尤其是分析与方法(AA)课程中，理解当自变量趋向无穷大（或负无穷大）时函数的行为至关重要。这个概念称为无穷远处的极限，它帮助我们识别水平渐近线、比较不同函数的增长速率，以及分析数学模型中的长期趋势。掌握无穷远处的极限为微积分中反常积分和级数收敛等内容奠定基础。

1. Introduction to Limits at Infinity | 无穷远处极限简介

When we write limₓ→∞ f(x) = L, we mean that as x increases without bound, the values of f(x) approach the number L. Similarly, limₓ→-∞ f(x) = M describes the behaviour as x becomes arbitrarily negative. These limits, if they exist, reveal the eventual tendency of a function.

当我们写 limₓ→∞ f(x) = L 时，意味着随着 x 无限增大，f(x) 的值趋近于某个数 L。类似地，limₓ→-∞ f(x) = M 描述 x 趋向负无穷时的行为。如果这些极限存在，它们揭示了函数的终态趋势。

For example, the function f(x) = 1/x becomes smaller and smaller as x grows, tending to 0. Hence, limₓ→∞ (1/x) = 0. This simple limit is fundamental and appears in many IB problems.

例如，函数 f(x)=1/x 随着 x 增大变得越来越小，趋于 0。因此 limₓ→∞ (1/x)=0。这个简单的极限是许多IB问题的基础。

2. Formal Definition and Notation | 正式定义与符号

Precisely, limₓ→∞ f(x) = L means that for every ε > 0, there exists a number N such that |f(x) – L| < ε whenever x > N. This epsilon definition formalizes the idea of ‘approaching’. In IB exams, you are not required to reproduce the formal definition, but understanding it can deepen your insight into why certain limits hold.

严格地说，limₓ→∞ f(x) = L 意味着对于任意 ε > 0，存在数 N，使得当 x > N 时，|f(x) – L| < ε。这个 ε 定义将“趋近”的概念形式化。在 IB 考试中，你不需要复述正式定义，但理解它可以加深你对为何某些极限成立的洞察力。

The notation limₓ→∞ f(x) = ∞ is also used, but this indicates that the function grows without bound rather than approaching a finite number. It is still described as a limit being infinite, though strictly it means the limit does not exist as a finite number.

符号 limₓ→∞ f(x) = ∞ 也常被使用，但这表示函数无限增长而不是趋近于有限数。虽然被称为无穷极限，严格来说它意味着极限不作为有限数存在。

3. Horizontal Asymptotes | 水平渐近线

If limₓ→∞ f(x) = L or limₓ→-∞ f(x) = L, the line y = L is called a horizontal asymptote of the graph of f. A function may have one, two, or no horizontal asymptotes. For rational functions, comparing degrees of numerator and denominator determines horizontal asymptotes.

如果 limₓ→∞ f(x) = L 或 limₓ→-∞ f(x) = L，则直线 y = L 称为函数 f 图像的水平渐近线。一个函数可能有一条、两条或没有水平渐近线。对于有理函数，比较分子与分母的次数可确定水平渐近线。

For instance, f(x) = (2x² + 1)/(3x² – 5) has horizontal asymptote y = 2/3, because limₓ→∞ f(x) = 2/3 and limₓ→-∞ f(x) = 2/3. In contrast, f(x) = (x² + 1)/(x + 1) has no horizontal asymptote because the degree of the numerator exceeds that of the denominator, so the limit is infinite.

例如 f(x) = (2x² + 1)/(3x² – 5) 有一条水平渐近线 y=2/3，因为 limₓ→∞ f(x)=2/3 且 limₓ→-∞ f(x)=2/3。相反，f(x) = (x² + 1)/(x + 1) 没有水平渐近线，因为分子次数高于分母次数，极限为无穷大。

4. Limits of Polynomial and Rational Functions | 多项式与有理函数的极限

For a polynomial p(x) = aₙxⁿ + … + a₀, the term of highest degree dominates as x → ±∞. Therefore, limₓ→∞ p(x) is ∞ or -∞ depending on the sign of the leading coefficient and whether the degree is even or odd. For example, limₓ→∞ (-2x³ + 5x) = -∞.

对于多项式 p(x)=aₙxⁿ+…+a₀，最高次项在 x → ±∞ 时占主导地位。因此，limₓ→∞ p(x) 取决于首项系数的符号以及次数是偶数还是奇数，得到 ∞ 或 -∞。例如，limₓ→∞ (-2x³ + 5x) = -∞。

For rational functions R(x) = p(x)/q(x), the limit at infinity is governed by the degrees of p and q. If deg p < deg q, limit is 0. If deg p = deg q, the limit is the ratio of leading coefficients. If deg p > deg q, the limit is ±∞ depending on signs and degree difference; no horizontal asymptote exists, but an oblique asymptote may appear.

对于有理函数 R(x)=p(x)/q(x)，无穷远处的极限由 p 和 q 的次数决定。如果 deg p < deg q，极限为 0。如果 deg p = deg q，极限为首项系数之比。如果 deg p > deg q，极限为 ±∞（取决于符号和次数差）；没有水平渐近线，但可能存在斜渐近线。

An illustrative case: limₓ→∞ (3x+2)/(x²-4) = 0, because the denominator degree is larger. Meanwhile, limₓ→∞ (4x⁵-3x²)/(2x⁵+7) = 4/2 = 2.

一个说明性例子：limₓ→∞ (3x+2)/(x²-4)=0，因为分母次数更高。而 limₓ→∞ (4x⁵-3x²)/(2x⁵+7)=4/2=2。

5. Technique: Dividing by Highest Power | 技巧：除以最高次幂

To evaluate limits of rational functions at infinity algebraically, dividing numerator and denominator by the highest power of x present in the denominator is a powerful method. For instance, to find limₓ→∞ (5x³ – x)/(2x³ + x²), divide numerator and denominator by x³: (5 – 1/x²)/(2 + 1/x). As x → ∞, 1/x and 1/x² approach 0, giving limit 5/2.

代数上求有理函数在无穷远处的极限时，分子分母同除以分母中 x 的最高次幂是一种有效方法。例如，求 limₓ→∞ (5x³ – x)/(2x³ + x²)，将分子分母同除以 x³ 得：(5 – 1/x²)/(2 + 1/x)。当 x → ∞ 时，1/x 和 1/x² 趋于 0，极限为 5/2。

This technique also works for expressions involving radicals. For limₓ→∞ √(4x² + x)/(2x + 1), factor out x² inside the radical: √(x²(4 + 1/x)) = |x|√(4 + 1/x). Since x → ∞, |x| = x, the expression becomes x√(4 + 1/x)/(2x + 1). Dividing numerator and denominator by x yields √(4 + 1/x)/(2 + 1/x) → √4/2 =

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）