The most important limit in Calculus // Geometric Proof & Applications

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The limit of sin(x)/x as x goes to zero is perhaps the most important limit in Calculus and it is just one! This is equivalent to the small angle approximation that sin(x) is approximately just equal to x when x is sufficiently small. In this video we will begin with an application from astronomy using angular distance to estimate the size of objects in the universe. Then we will focus on the geometric proof of the limit of sin(x)/x which follows from some simple trigonometry. It is important to proof this geometrically because we can’t just use L’Hopitals rule as that requires knowing that the derivative of sin(x) is just cos(x). As we will see, that proof however relies on this very limit!

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0:00 Limit of sin(x)/x as x goes to 0
0:56 Astronomy Application
2:49 Visual Intuition using Maple Learn
3:47 Geometric Proof
8:16 Proof that the derivative of sin(x) is cos(x)

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