📚 Two Important Limits in A-Level Maths | A-Level 数学中的两个重要极限
In AQA A-Level Mathematics, the two important trigonometric limits are essential tools for proving the derivatives of sin x and cos x from first principles. They are small but powerful results that appear in many limit problems and underpin the calculus of trigonometric functions.
在 AQA A-Level 数学中，两个重要三角极限是用第一性原理证明 sin x 和 cos x 导数的重要工具。这两个结果看似简单，但在许多极限问题中都非常有用，并且是三角函数微积分的基础。
1. What Are the Two Important Limits? | 两个重要极限是什么？
The two limits are written as follows, where x is measured in radians.
这两个极限如下所示，其中 x 以弧度为单位。
lim (x→0) (sin x)/x = 1
lim (x→0) (1 − cos x)/x = 0
The first limit says that when x is very close to 0, sin x behaves almost exactly like x. The second says that 1 − cos x tends to 0 much faster than x does.
第一个极限说明，当 x 非常接近 0 时，sin x 的行为几乎与 x 完全相同。第二个极限说明，1 − cos x 趋向 0 的速度比 x 快得多。
2. Radian Measure: Why It Is Essential | 弧度制：为什么必不可少
These limits are only true when angles are measured in radians. If x were in degrees, the first limit would become π/180 instead of 1, because the derivative of sin x would be scaled by the conversion factor.
这些极限只有在角度以弧度为单位时才成立。如果 x 以度为单位，第一个极限就会变成 π/180，而不是 1，因为 sin x 的导数会受到单位换算因子的影响。
In AQA exam questions, you should always check that any trigonometric limit is expressed in radians before applying the standard results. Some questions may try to catch you out by using degrees or by asking you to explain why the limit changes.
在 AQA 考试题中，在应用这些标准结果之前，你应始终确认任何三角极限都是用弧度表示的。有些问题可能会故意使用角度制，或要求你解释极限为什么会发生变化。
3. First Limit: lim (x→0) sin x / x = 1 | 第一个极限：lim (x→0) sin x / x = 1
Consider the numerical behaviour of sin x / x as x approaches 0. The table shows that the ratio gets closer and closer to 1.
观察当 x 趋近于 0 时 sin x / x 的数值变化。下表说明该比值越来越接近 1。
| x (radians) | sin x / x |
|---|---|
| 0.1 | 0.998334... |
| 0.01 | 0.999983... |
| 0.001 | 0.999999... |
The values suggest that the ratio becomes closer to 1 as x gets smaller, but a numerical table alone is not a proof. A rigorous AQA answer should include the geometric squeezing argument shown in the next section.
这些数值表明，随着 x 变小，比值越来越接近 1，但仅仅依靠数值表并不能构成证明。在 AQA 考试中，严谨的解答应包含下一节展示的几何夹逼论证。
4. Geometric Proof of the First Limit | 第一个极限的几何证明
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