Using the Sandwich Theorem

If √(5 −2x²) ≤ f(x) ≤ √(5−x²) for −1 ≤ x ≤ 1, find limx→0 f(x).

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Understand the Sandwich Theorem: It states that if a function f(x) is squeezed between two other functions g(x) and h(x) such that g(x) ≤ f(x) ≤ h(x) for all x in some interval, and if the limits of g(x) and h(x) as x approaches a certain value are equal, then the limit of f(x) as x approaches that value is the same.

Identify the functions involved: Here, we have g(x) = √(5 − 2x²) and h(x) = √(5 − x²) with f(x) squeezed between them, i.e., √(5 − 2x²) ≤ f(x) ≤ √(5 − x²).

Determine the limits of g(x) and h(x) as x approaches 0: Calculate limx→0 √(5 − 2x²) and limx→0 √(5 − x²).

Calculate limx→0 √(5 − 2x²): Substitute x = 0 into the expression to find the limit.

Calculate limx→0 √(5 − x²): Substitute x = 0 into the expression to find the limit. Since both limits are equal, apply the Sandwich Theorem to conclude that limx→0 f(x) is equal to this common limit.

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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 f(x) is 'squeezed' between two other functions g(x) and h(x) such that g(x) ≤ f(x) ≤ h(x) for all x in an interval, and if the limits of g(x) and h(x) as x approaches a certain value are equal, then the limit of f(x) as x approaches that value 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 lim x→0 f(x), which means we need to evaluate the behavior of f(x) as x gets closer to 0, using the bounds provided by the functions √(5 − 2x²) and √(5 − x²).

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. In this problem, since the functions √(5 − 2x²) and √(5 − x²) are continuous over the interval [-1, 1], we can confidently apply the Sandwich Theorem to find the limit of f(x) as x approaches 0.

Using the Sandwich TheoremIf √(5 −2x²) ≤ f(x) ≤ √(5−x²) for −1 ≤ x ≤ 1, find limx→0 f(x).