In the Kingdom of Rimonn there are three rivers. In this lesson, learn how these waterways demonstrate the power of the squeeze theorem for finding the limits of functions.
Introduction to the Squeeze Theorem
Depiction of the three primary rivers and the village of Moe
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.
Understanding the Squeeze Theorem
Understanding the squeeze theorem
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.
Use the product property to divide the limit into two limits
Squeeze Theorem in Practice
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.
In this example, use -1/x and 1/x to squeeze sin(x)/x
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.
Squeeze Theorem - Applications
The Squeeze Theorem is a technique to evaluate the limit of a function by finding two functions that bound the given function below and above and the two functions approach the same finite value when the limits are evaluated.
Squeeze Theorem is usually used when we have sine or cosine terms because they are bounded by -1 and 1.
Application - Limits in Two Variables
For example, the limit of a function of two variables at a point (a, b) is the finite value, L, such that, for any disk centered at (a, b), whose radius goes to zero (so the disk approaches its center), the function approaches the unique value L:
A disk can be represented in polar coordinates as:
So a two-dimensional limit can be written as a one-dimensional limit.
Usually, the limit in polar coordinates is evaluated with the Squeeze Theorem.
To evaluate the limits below, convert them to polar coordinates and evaluate.
1. Using the polar coordinates centered at the origin and radius r:
Because the cosine term is bounded by -1 and 1, the square of cosine is bounded by 0 and 1, so:
And by the Squeeze Theorem, the limit is zero.
2. Using the polar coordinates centered at the point, (1, 0), and radius, r:
the limit is
which does not exist
3. Using the polar coordinates centered at the origin and radius r, we obtain:
This does not exist because the limit depends on theta, so the limit will not be unique, it will be different for different values of theta.