Solve Difficult Limits With The Squeeze Theorem In Calculus

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, also known as the sandwich theorem, is a fundamental theorem in calculus that is used to find the limit of a function. It states that if two functions, let’s say f(x) and g(x), approach the same limit L as x approaches some value a, and if we have another function, let’s say h(x), which is squeezed between f(x) and g(x) for all values of x near a (excluding perhaps at a), then h(x) must also approach the same limit L as x approaches a.

Mathematically, the squeeze theorem can be represented as:

If f(x) ≤ h(x) ≤ g(x) for all x near a (excluding perhaps at a) and lim f(x) = lim g(x) = L as x → a, then lim h(x) = L as x → a.

The squeeze theorem is useful in evaluating limits that are difficult to compute directly. It is often applied to trigonometric functions or exponential functions. In order to use the squeeze theorem, one needs to find two simpler functions (usually constants or simpler functions with known limits) that are greater than or less than the given function and approach the same limit L as the given function.

Overall, the squeeze theorem is a powerful tool for evaluating limits and is essential to many of the more advanced concepts in calculus.

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