Introduction to limits
In many calculus textbooks, introduction to limits often starts with examples of limits such as [math]\displaystyle{ \lim_{x\to 2} x^2 }[/math] with invitation to evaluate the function, [math]\displaystyle{ x^2 }[/math] in this example, at points near [math]\displaystyle{ a }[/math] (so [math]\displaystyle{ x = }[/math] 2.1, 2.01, 1.9, 1.99 and so on). However, this type of examples is not particularly engaging and in fact may raise some questions as of why one needs to even consider limits in the first place. It would be more beneficial to introduces limits using examples in which the answer is at least not obvious. Another classical example is [math]\displaystyle{ \lim_{x\to 0} \frac{\sin x}{x} }[/math]. There are two problems with this one: it appears out of nowhere (i.e. as it presented is not motivated) and it is non-trivial to compute. The way around it is outlined below. Suppose you want to determine wether your planet is spherical or flat. You can draw a circle of some radius [math]\displaystyle{ r }[/math] and compare the circumference of this circle to its radius and analyze what happens with the circumference divided by the radius.
Plane
Let [math]\displaystyle{ C_r }[/math] denote the circumference of the circle of radius [math]\displaystyle{ r }[/math]. Then [math]\displaystyle{ C_r= 2\pi r }[/math] and therefore [math]\displaystyle{ \frac{C_r}{r}= 2\pi }[/math]; in particular this ratio does not depend on [math]\displaystyle{ r }[/math]
Sphere
Note that we measure the radius of the circle along the surface of the planet as the shortest distance from the centre to the circle. Let [math]\displaystyle{ O }[/math] be the centre of the circle on the surface of the sphere, [math]\displaystyle{ S }[/math] be the centre of the sphere, and [math]\displaystyle{ R }[/math] be the radius of the sphere. Let also [math]\displaystyle{ A }[/math] denote the "real" centre of the circle, i.e. the point on the segment [math]\displaystyle{ OS }[/math] and in the plane containing our circle. Then [math]\displaystyle{ C_r = 2\pi {\color{red} a} }[/math] where [math]\displaystyle{ a=r\sin \theta }[/math] (see the diagram). Also, it is easy to see that [math]\displaystyle{ r = R\theta }[/math]. Therefore [math]\displaystyle{ \frac{C_r}{r} = \frac{2\pi a}{r} = \frac{2 \pi R\sin\theta}{R\theta} = 2\pi \frac{\sin \theta }{\theta} }[/math]