Using the Sandwich Theorem

If 2−x² ≤ g(x) ≤ 2cosx for all x, find limx→0 g(x).

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Identify the functions involved in the inequality: f(x) = 2 - x² and h(x) = 2cos(x). We know that f(x) ≤ g(x) ≤ h(x) for all x.

Evaluate the limit of f(x) as x approaches 0: lim(x→0) (2 - x²). Since x² approaches 0 as x approaches 0, the limit of f(x) is 2.

Evaluate the limit of h(x) as x approaches 0: lim(x→0) 2cos(x). Since cos(0) = 1, the limit of h(x) is 2.

Apply the Sandwich Theorem (also known as the Squeeze Theorem), which states that if f(x) ≤ g(x) ≤ h(x) and lim(x→a) f(x) = lim(x→a) h(x) = L, then lim(x→a) g(x) = L.

Conclude that since both limits of f(x) and h(x) as x approaches 0 are equal to 2, by the Sandwich Theorem, lim(x→0) g(x) = 2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sandwich Theorem

The Sandwich Theorem, also known as the Squeeze Theorem, states that if a function g(x) is bounded by two other functions f(x) and h(x) such that f(x) ≤ g(x) ≤ h(x) for all x in an interval, and if the limits of f(x) and h(x) as x approaches a point are equal, then the limit of g(x) as x approaches that point is also equal to that limit.

Limit of a Function

The limit of a function describes the value that the function approaches as the input approaches a certain point. In this context, we are interested in finding the limit of g(x) as x approaches 0, which requires evaluating the behavior of g(x) near that point based on the bounding functions.

Bounding Functions

Bounding functions are the functions that enclose another function within a specific interval. In this case, 2−x² and 2cosx serve as the bounding functions for g(x). Understanding their limits as x approaches 0 is crucial for applying the Sandwich Theorem to find the limit of g(x).

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