Textbook Question
If f(1)=5, must limx→1 f(x) exist? If it does, then must limx→1 f(x)=5? Can we conclude anything about limx→1 f(x)? Explain.
1. Limits and Continuity
Introduction to Limits
2:26 minutesProblem 2.2.63
Textbook Question
Using the Sandwich Theorem
If √(5 −2x²) ≤ f(x) ≤ √(5−x²) for −1 ≤ x ≤ 1, find limx→0 f(x).
Verified step by step guidance1
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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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 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.
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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).
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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.
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