## The Squeeze Theorem

*Interesting fact*: The textbooks in China (for the international audience) and many other countries translate this into "the sandwich theorem", which is an even more vivid way of describing the essence of this principle.

This principle simply says, if f(x)<=g(x)<=h(x) for all x, then for any value a, the limit of f(x), g(x), h(x) as x approach a follows the rule:

lim f(x)<=lim g(x)<=lim h(x). This, if we take a moment of thought, is obvious.

Consider the three curves in the plane. The curve of f(x) always stays above the curve of g(x), and the curve of g(x) always stay above the curve of h(x). And the limit?

Think of it as a prison. h(x) and f(x) are the guards, guarding g(x) in between. of course, g(x) could not go outside of the bounds defined by the two guards. (assuming ideal "unbribable" guards, that sadly, only exist in the kingdom of mathematics)

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