Master The Squeeze Theorem: A Calculus Method For Finding Limits Of Functions

The squeeze theorem

If f(x) ≤ g(x) ≤ h(x) for all x near a and limx→a f(x) = limx→a h(x) = L, thenlimx→a g(x) = L

The squeeze theorem is also known as the sandwich theorem or the pinching theorem. It is a mathematical principle used in calculus to prove limits of functions. The theorem is used when we need to find the limit of a function that cannot be evaluated directly by substitution or other methods. It involves comparing the function we want to find the limit of with two other functions which are known to have the same limit.

Mathematically, if f(x) <= g(x) <= h(x) for all x in some interval except at a, where both f(x) and h(x) approach the same limit L as x approaches a, then g(x) also approaches L as x approaches a. Symbolically, we can write it as follows: If, for all x in some interval except at a, we have: f(x) <= g(x) <= h(x) and lim x->a f(x) = L = lim x->a h(x) , then, lim x->a g(x) = L.

The theorem works by bounding the target function between two other functions and proving that as the bounds converge to the same limit, the original function also converges to that limit.

The squeeze theorem is particularly useful in calculus when evaluating limits of functions that may not have a straightforward method of evaluation. It is often used to prove the limit of a function using known limits of two other functions.

Overall, the squeeze theorem is a powerful tool in the study and evaluation of limits, and it can be used to prove limits of functions that may not be easily evaluated directly.

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