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The sandwich rule, also known as the squeeze theorem, is a method used to find the limit of a function when it is sandwiched between two other functions. It states that if two functions, g(x) and h(x), approach the same limit as x approaches a certain value, and the function f(x) is bounded between them, then the limit of f(x) as x approaches that value is equal to the limit of g(x) and h(x).
The sandwich rule should be used when evaluating the limit of a function that is not defined at a certain point or is approaching infinity. It can also be used when the limit of a function is not easily determined using other methods such as direct substitution or algebraic manipulation.
To use the sandwich rule, you must first identify the two functions, g(x) and h(x), that have the same limit as x approaches the value in question. Then, you must show that the function f(x) is bounded between g(x) and h(x). This can be done by finding the limits of g(x) and h(x) and showing that they approach the same value, or by using algebraic manipulation to show that f(x) is always between g(x) and h(x).
Yes, there are some limitations to the sandwich rule. It can only be used when the two functions, g(x) and h(x), have the same limit as x approaches the value in question. If the limit of these functions is different, then the sandwich rule cannot be applied. Additionally, the function f(x) must be bounded between g(x) and h(x) for the rule to be valid.
No, the sandwich rule is most commonly used for evaluating the limits of trigonometric, exponential, and logarithmic functions. It can also be used for rational functions and piecewise functions. However, it may not be applicable for more complex functions with multiple variables or undefined points.
