Psychedelic Math Rock

I love learning about the sort of things most people don't want to know about, because their perception of, say, rocket science, is that it must be far too complicated for the average person to begin to comprehend. But I am the sort of person who needs to know how things work, which is probably what led to me buying a book on quantum physics a couple of years ago. However, even I have to admit that calculus takes some figuring out. Take this as an example;

Squeezing Theorem.

If f, g and h are functions such that

f(rhubarb) <= g(rhubarb) <= h(rhubarb)

for all values of rhubarb in some open interval containing custard and if lim_{rhubarb→custard} f(rhubarb) = lim_{rhubarb→custard} h(rhubarb) = L then

lim_{rhubarb→custard} g(rhubarb) = L

How to use the squeezing theorem?
Example 1: Find the limit
lim_rhubarb→0 rhubarb ² cos(1/rhubarb)
Solution to Example 1:

As rhubarb approaches 0, 1 / rhubarb becomes very large in absolute value and cos(1 / rhubarb) becomes highly oscillatory. However cos(1 / rhubarb) takes values within the interval [-1,1] which is the range of cos rhubarb, hence

-1 ≤ cos (1/rhubarb) ≤ 1

Multiply all terms of the above inequality by rhubarb ² (rhubarb not equal to 0)

- rhubarb ² ≤ rhubarb ² cos (1/rhubarb) ≤ rhubarb ²

The above inequality holds for any value of rhubarb except 0 where rhubarb ² cos (1/rhubarb) in undefined. As rhubarb approaches 0 both - rhubarb ² and rhubarb ² approach 0 and according to the squeezing theorem we obtain

lim_rhubarb→0 rhubarb ² cos(1/rhubarb) = 0
Example 2: Find the limit lim_rhubarb→0 sin rhubarb / rhubarb
Solution to Example 2:

Assume that 0 < rhubarb < Pi/2 and let us us consider the unit circle, shown below, and a sector ChickenFriedRice with central angle rhubarb where rhubarb is in standard position. Fried is a point on the unit circle and ChickenSoup is tangent to the circle at Chicken.

Point Rice is a point on the unit circle (radius = 1)and has coordinates (cos rhubarb, sin rhubarb). Let us find the areas of triangle ChickenFriedRice, sector ChickenFriedRice and triangle

area of triangle ChickenFriedRice = (1/2)*(base)*(height)

= (1/2)*(1)*(y coordinate of point C) = (1/2)(sin rhubarb)

Note: we have used base = radius = 1

area of sector ChickenFriedRice = (1/2)*(rhubarb)*(radius) ²

= (1/2) (1) ² rhubarb = (1/2) rhubarb

area of triangle ChickenFriedSoup = (1/2)*(base)*(height)

= (1/2)*(1)*(tan rhubarb) = (1/2) tan rhubarb Comparing the three areas, we can write the inequality

area of triangle ChickenFriedRice < area of sector ChickenFriedRice < area of triangle ChickenFriedSoup Substitute the areas in the above inequality by their expressions obtained above.

(1/2)(sin rhubarb) < (1/2) rhubarb < (1/2) tan rhubarb Multiply all terms by 2 / sin rhubarb gives

1 < rhubarb / sin rhubarb < 1 / cos rhubarb Take the reciprocal and reverse the two inequality symbols in the double inequality

1 > sin rhubarb / rhubarb > cos rhubarb Which the same as

cos rhubarb < sin rhubarb / rhubarb < 1 It can be shown that the above inequality hols for -Pi/ 2 < rhubarb < 0 so the the above inequality hold for all rhubarb except rhubarb = 0 where sin rhubarb / rhubarb is undefined. Since

lim_rhubarb→0 cos rhubarb = 1 and

lim_rhubarb→0 1 = 1 , we can apply the squeezing theorem to obtain

lim_rhubarb→0 sin rhubarb / rhubarb = 1 This result is very important and will be used to find other limits of trigonometric functions and derivatives

See what I mean? That makes no sense, whatsoever! ChickenFriedSoup? What the hell is that?