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Squeeze theorem
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The squeeze theorem is a useful way to find a limit in certain specific situations. In this article, we will use a simple example to explain how the squeeze theorem works, and then go on to prove the theorem.
Example — x² \sin (1/x)
As a first example, we will use the squeeze theorem to find:
Where:
The function is shown here:
The problem here is that we cannot evaluate or find the limit of sin (1/x) at zero because the argument 1/x goes to infinity, so the function oscillates infinitely many times as it approaches 0.
What can we do? Well, we can observe that the value of sin (1/x) is always in the range [-1, 1] for any value of x. Even though its value oscillates infinitely many times as we move towards zero, it can never go outside that range. In other words:
This alone doesn’t help us find the limit, because although the function is bounded, it is the oscillations that cause the problem. In order for some function u(x) to have a limit L and x approaches some value a, then as x gets very close to a we expect u(x) to get very close to L.
But in the case of sin (1/x), as x approaches 0, this condition is not true. If we choose…