Do you know that some limits from the limit theory are considered “wonderful”? Actually, I’ve never met such a term in English literature, but it’s widely known in post-soviet countries. In Belarusian, it’s written as “грунтоўныя ліміты” [hruntoŭnyja limity], which can be translated in this context as “fundamental” or “wonderful” limits, and these limits are even studied in universities on this topic.
So what are these unusual limits that are called “wonderful”, and why are they so special? There are two such limits, but we’ll consider the first one in this article. And here it is:
Let’s see why this equation is true. First of all, we consider the next triangles in the unit circle (AB = 1):
The area of the triangle ABC is 0.5 sin(x). The area of the arc CAB is 0.5 x, and the area of the triangle ABD is 0.5 tan(x). By inclusion, we get the next double inequity:
Let’s take the inverse of all the parts of the inequality. Thus we get:
At this step, we’ll multiply all the parts of the equation by sin(𝑥) and reduce it. Since tan(x) = sin(x)/cos(x), we have:
Let’s take the limit when x approaches 0. As we know, when x → 0, cos(x) → 1 and 1 → 1. Thus by the squeeze theorem we get, that sin(x)/x → 1 when x → 0:
And this is the proof of the first wonderful limit.
Why is this limit so wonderful? First, as you can see, it has really nice proof. You don’t need to define any functions or choose the right value of a parameter to make everything work. It comes from the definition of sine. The second reason is that you can easily find other limits using this one. For example, we can prove, that tan(x)/x approaches 1 when x approaches 0:
I hope you enjoyed reading this article. See you in the second part!
