- sinxx.png (8.28 KiB) Viewed 150 times
In this picture, there're two right triangles and a circular sector whose radius is 1.
If we rotate these shapes by x-axis, then two right triangles become two cones and a sector becomes a spherical sector.
We can evaluate the limit of (1 - cos θ)/θ², θ → 0+ by comparing volume of each shape.
Volume of each shape can be evaulated by integrating areas of circular cross sections, but what would happen if we integrate volumes of spheres instead here?
Then two cones become two spherones and a spherical sector becomes something 4D shape.
Let's evaulate hypervolume of these 4D shapes:
- 극한.png (16.29 KiB) Viewed 149 times
Finally we can evaluate the limit of (x - sin x)/x³, x → 0+ by comparing these hypervolumes. Multiply 4/π to all sides then subsitute 2θ = x, divide all sides by x³, apply squeeze theorem.
For x → 0-, just substitute s = -x then you'll got same result of x → 0+.