Squeeze theorem

The squeeze theorem is an important idea in calculus. It helps us find the limit of a function by comparing it to two other functions. If a function is always between two other functions, and those two functions have the same limit, then the middle function must have that same limit too.

This theorem is often called the sandwich theorem because it’s like putting something in between two slices of bread. It was first used a long time ago by smart mathematicians like Archimedes and Eudoxus to help figure out the value of π. Later, Carl Friedrich Gauss helped put the idea into its modern form.

The squeeze theorem is very useful in mathematical analysis. It gives us a way to prove what the limit of a tricky function is, by using what we already know about simpler functions.

Statement

The squeeze theorem is a way to find the limit of a function by comparing it to two other functions. If a function is always between two other functions, and those two functions have the same limit at a certain point, then the middle function must also have that same limit at that point.

This theorem helps us figure out limits that might be tricky to calculate directly. For example, it can be used to show that certain complex expressions approach a simple value when they get very close to a specific point. It’s a useful tool in calculus for understanding how functions behave near particular values.