Limits at Infinity | 无穷处的极限

📚 Limits at Infinity | 无穷处的极限

The concept of a limit at infinity in IB Mathematics explores the behavior of a function \(f(x)\) as the independent variable \(x\) increases or decreases without bound. This is denoted as \(x \to \infty\) or \(x \to -\infty\). Instead of asking what happens at a finite point, we ask: where does the function eventually settle, if it settles at all?

IB 数学中，无穷处的极限研究的是当自变量 \(x\) 无限增大或无限减小时，函数 \(f(x)\) 的变化趋势，记作 \(x \to \infty\) 或 \(x \to -\infty\)。我们不再关注某一点上的取值，而是关注函数终将趋于何处，或者说它是否最终会稳定下来。

1. Polynomial Functions | 多项式函数

For polynomials, the leading term (the term with the highest power) completely governs the limit at infinity. Lower-degree terms become insignificant as \(|x|\) grows extremely large. This is often called “grabbing the biggest head” (抓大头) in Chinese mathematics culture.

对于多项式，最高次项（即幂指数最大的项）完全决定了无穷远处的极限。当 \(|x|\) 变得极大时，所有低次项的影响都微乎其微，这一点在中国学生中常被称为”抓大头”。

lim x→∞ (3x³ − 2x² + 5) = ∞

If the leading coefficient is positive, the function tends toward positive infinity. If the leading coefficient is negative, it tends toward negative infinity. Even-degree polynomials behave identically on both the positive and negative sides, while odd-degree polynomials will have opposite signs.

若最高次项的系数为正，函数趋向于正无穷；若系数为负，则趋向于负无穷。偶次多项式在 \(x\) 趋近于正、负无穷时的极限行为一致；而奇次多项式在两侧的极限则会符号相反。

2. Rational Functions: Degree Comparison | 有理函数：次数比较

For rational functions of the form \(P(x)/Q(x)\), where both are polynomials

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

更多咨询请联系16621398022（同微信）