Squeeze Theorem: Evaluating Limits Of Functions In Calculus | Sandwich Theorem Explained

Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) for all x ̸= 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 or the pinching theorem, is a theorem used in calculus to evaluate the limit of a function when the limit of two other functions whose values are always in between the function in question, approach a common limit.

The formal definition of the squeeze theorem is as follows: Suppose we have three functions f(x), g(x), and h(x) defined on an interval containing a point c, except possibly at c itself. If f(x) is always in between g(x) and h(x) in that interval, and if the limits of g(x) and h(x) as x approaches c are both L, then the limit of f(x) as x approaches c is also L. Symbolically, it can be written as:

if g(x) ≤ f(x) ≤ h(x) for all x in the interval (c – δ, c + δ), except possibly at c, and limx→c g(x) = limx→c h(x) = L,
then limx→c f(x) = L.

It is a powerful tool that is used to find limits of functions that may not have a common limit or may not be easily evaluated using other means. It is particularly useful when dealing with trigonometric functions, exponential functions, and logarithmic functions.

Some common applications of the squeeze theorem include evaluating limits involving trigonometric functions such as sin x/x and tangent functions such as tan x/x, as well as limits involving exponential functions such as e^x-1/x and logarithmic functions like ln(x+1)/(x+1).

In summary, the squeeze theorem helps us to determine the limit of a function by squeezing it between two other functions whose limits are known to be the same.

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