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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주요 개념

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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).

Using the Sandwich TheoremIf 2−x² ≤ g(x) ≤ 2cosx for all x, find limx→0 g(x).