📚 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 → ∞ or x → -∞. 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 → ∞ 或 x → -∞。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
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