Textbook Question
Which of the following statements about the function y=f(x) graphed here are true, and which are false?
a. limx→2 f(x) does not exist.
1. Limits and Continuity
Introduction to Limits
2:28 minutesProblem 2.2.65
Textbook Question
Using the Sandwich Theorem
a. It can be shown that the inequalities 1 − x²/ 6 < (x sin x) / (2−2cos x) < 1 hold for all values of x close to zero (except for x = 0). What, if anything, does this tell you about limx→0 (x sin x) / (2 − 2cos x)?
Give reasons for your answer.
[Technology Exercise] b. Graph y = 1 − (x²/6), y=(x sinx)/(2 − 2cos x), and y = 1 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.
Verified step by step guidance1
Step 1: Understand the Sandwich Theorem, also known as the Squeeze Theorem. It states that if you have three functions f(x), g(x), and h(x) such that f(x) ≤ g(x) ≤ h(x) for all x in some interval around a point (except possibly at the point itself), and if limx→c f(x) = limx→c h(x) = L, then limx→c g(x) = L.
Step 2: Apply the Sandwich Theorem to the given inequalities. We have 1 − x²/6 < (x sin x) / (2 − 2cos x) < 1 for values of x close to zero (except x = 0). We need to find the limits of the bounding functions as x approaches 0.
Step 3: Calculate the limit of the lower bound function as x approaches 0. The function is 1 − x²/6. As x approaches 0, x² approaches 0, so the limit of 1 − x²/6 as x approaches 0 is 1.
Step 4: Calculate the limit of the upper bound function as x approaches 0. The function is simply 1, which is constant. Therefore, the limit of 1 as x approaches 0 is also 1.
Step 5: Conclude using the Sandwich Theorem. Since both the lower bound and upper bound functions approach the same limit of 1 as x approaches 0, by the Sandwich Theorem, the limit of (x sin x) / (2 − 2cos x) as x approaches 0 is also 1.
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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 is squeezed between two other functions that both converge to the same limit at a certain point, then the squeezed function must also converge to that limit at that point. This theorem is particularly useful in evaluating limits of functions that are difficult to analyze directly.
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Limits
In calculus, a limit describes the behavior of a function as its input approaches a certain value. It is a fundamental concept that helps in understanding continuity, derivatives, and integrals. The limit can be finite or infinite, and it is essential for determining the value of functions at points where they may not be explicitly defined.
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Graphical Analysis
Graphical analysis involves examining the behavior of functions through their graphs to gain insights into their limits, continuity, and overall behavior. By plotting functions, one can visually assess how they interact, especially near critical points like limits. This method is particularly useful for understanding complex functions and their asymptotic behavior as they approach specific values.
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