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Math 104: Calculus13 chapters | 105 lessons
Erin has taught math and science from grade school up to the post-graduate level. She holds a Ph.D. in Chemical Engineering.
Welcome to the Kingdom of Rimonn! Now in the Kingdom of Rimonn, we have three primary rivers. We have the River Euler. We have the River Newton. And we have the River Tiny. We don't exactly know where Tiny goes, but we know he starts out in the hills and he ends in the sea. We know a few things about the rivers in the Kingdom of Rimonn. We know that Euler is always north of the River Newton. We know that Tiny is always north of Newton, but south of Euler. So we know that Tiny basically is always between Newton and Euler, we just don't know exactly where he goes.
We also know that Euler and Newton meet up. They get very close in a village called Moe. So because Tiny is surrounded by Newton and Euler throughout the entire length of the river, we know that Tiny also has to meet up in the town of Moe. Because we know that Tiny doesn't cross Euler or Newton, and since they meet up at Moe, Tiny must also meet up at Moe.
This principle is known as the squeeze theorem in calculus. Some people call it the sandwich theorem, but I like the term squeeze.
Now let's consider the village of Moe, and let's zoom in really close where Euler and Newton meet up. I can say that the limit, as we approach Moe, of Euler is this point here. Let's call it the town square. And the limit, as we approach Moe, of Newton is also the town square. Because Euler is always north of Tiny and Tiny is always north of Newton, I can write that the limit, as we approach Moe, of Tiny is also the town square.
So let's make this really formal. If the function g(x) is less than or equal to f(x), which is less than or equal to h(x), and the limit, as we approach some number, of g(x) equals the limit, as we approach that same number, of h(x), then we've squeezed f(x) such that the limit, as we approach the same number, of f(x) is equal to both the limit of both g and h. In this case, h is like Euler, g is like Newton and f is like Tiny, and f is squeezed in here. So the limit as we approach Moe is that town square.
The best example of the squeeze theorem in practice is looking at the limit as x gets really big of sin(x)/x. I know from the properties of limits that I can write this as the limit, as x goes to infinity, of sin(x) divided by the limit, as x goes to infinity, of x, as long as x exists as this gets really, really big. But I can also write this as the limit, as x goes to infinity, of sin(x) * 1/x. I can use multiplication, the product property, to divide this into two limits.
Now to use the squeeze theorem, we need to look at what possible functions might surround this sin(x)/x. What will always be bigger and what will always be smaller? Well, sin(x) is always going to be between -1 and 1. So perhaps I can write that sin(x)/x will always be greater than or equal to -1/x. And sin(x) will always be less than or equal to 1/x. So maybe we can use -1/x and 1/x to squeeze sin(x)/x. So what happens to -1/x and 1/x as x gets really big? Well, as x gets really big, -1/x gets really close to zero. So the limit, as x gets really big, of -1/x is 0. Similarly, if we look at 1/x, the limit, as x goes to infinity, of 1/x is also zero. What we have here is that as we get very large, sin(x)/x is surrounded by things that are going to zero. So the limit, as x gets really large, of sin(x)/x must be zero.
So to recap, when you're thinking about the squeeze theorem, think of the kingdom of Rimonn, and think about what has to happen to the rivers of Newton, Euler and Tiny as we approach the village of Moe. Because Euler and Newton are going through the town square of Moe, Tiny must also. This is the same thing as saying if g(x) is less than or equal to f(x), which is less than or equal to h(x), and the limit, as we approach some number, of g(x) equals the limit, as we approach that number, of h(x), then f(x) must also approach that number.
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