How To Find Limits With The Squeeze Theorem In Math

Squeeze theory (AKA sandwich theory or pinch theory) for limit functions

f(x) ≤ g(x). when x is near a (but possibly not a) thenlim (x → a) f(x) ≤ lim (x → a) g(x) also is. f(x) ≤ (g(x) ≤ h(x). when x is near a (but possible not a) andlim (x → a) f(x) = lim (x → a) h(x) = L. thenlim (x → a) g(x) = L

The squeeze theory (also known as the sandwich theory or pinch theory) is a powerful tool for determining the limit function of a sequence or function. It is used when we have two other functions that squeeze the original function, such that the two functions approach the same value as the original function as they move towards a specific point.

In mathematics, suppose we have a sequence or function f(x) that we are trying to find the limit for as x approaches a certain value, say c. If we can find two other functions g(x) and h(x) such that g(x) ≤ f(x) ≤ h(x), and both g(x) and h(x) approach the same limit as x approaches c, then we can say that f(x) also approaches that limit.

The formal statement of the squeeze theorem is as follows:

If g(x) ≤ f(x) ≤ h(x) for all x in a deleted neighborhood of c, and both g(x) and h(x) approach L as x approaches c, then f(x) also approaches L as x approaches c.

In other words, if we can squeeze f(x) between two functions g(x) and h(x) that have the same limit at c, then f(x) must also approach that limit.

The squeeze theorem is often used in cases where we have functions or sequences that would be difficult or impossible to evaluate directly using other methods, such as L’Hopital’s rule or algebraic manipulations. By finding two simpler functions that sandwich the original function, we can often quickly and easily determine its limit.

Overall, the squeeze theory can be a very useful tool for finding the limit of a function or sequence, and it is worth understanding and applying in a variety of mathematical contexts.

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