The sandwich theorem (also called the squeeze theorem)

Not another post on limits…

Too late.

The sandwich theorem, known as the squeeze theorem in certain curriculums, is a fairly important theorem on limits.

In simple terms, it states that if ƒ₁ ≤ ƒ₂ ≤ ƒ₃ are three functions and the limit as ƒ₁ and ƒ₃ tends to a number is the same, then the limit as ƒ₂ tends to that number will also be the same.

The following graph illustrates this quite clearly. It also makes clear why this theorem is described using words like sandwich and squeeze.

So, now that you’ve got a hang of this thing, let’s take an example and analyse it non-mathematically. All you have to do is find two functions the given function is in between. Then find the limit you need to find, but of those two functions, and if they’re the same thing at the given point, then any function in between them will have the same limit at that point.

Go on, have fun with this, because it’s a really powerful, and more than that, elegant way of doing things.