Textbook Question
Which of the following statements about the function y=f(x) graphed here are true, and which are false?
b. limx→2 f(x)=2
1. Limits and Continuity
Introduction to Limits
2:04 minutesProblem 2.2.66
Textbook Question
Using the Sandwich Theorem
a. Suppose that the inequalities 1/2 − x² / 24 < (1 − cos x)/ x² < 1/2 hold for values of x close to zero, except for x = 0 itself. (They do, as you will see in Section 9.9.) What, if anything, does this tell you about limx→0 (1 −cos x)/ x²?
Give reasons for your answer.
[Technology Exercise] b. Graph the equations y=(1/2) − (x²/24), y = (1 - cos x) / x², and y = 1/2 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 the limits of f(x) and h(x) as x approaches that point are equal, then the limit of g(x) as x approaches that point is the same.
Step 2: Apply the Sandwich Theorem to the given inequalities. We have 1/2 - x²/24 < (1 - cos x)/x² < 1/2 for values of x close to zero. 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 f(x) = 1/2 - x²/24. As x approaches 0, x²/24 approaches 0, so the limit of f(x) is 1/2.
Step 4: Calculate the limit of the upper bound function as x approaches 0. The function is h(x) = 1/2, which is constant. Therefore, the limit of h(x) as x approaches 0 is also 1/2.
Step 5: Conclude using the Sandwich Theorem. Since both the lower bound and upper bound functions have the same limit of 1/2 as x approaches 0, by the Sandwich Theorem, the limit of (1 - cos x)/x² as x approaches 0 is also 1/2.
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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 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 that are difficult to compute directly.
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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 the limit of (1 - cos x) / x² as x approaches 0. Understanding limits is fundamental in calculus, as it lays the groundwork for concepts such as continuity and derivatives.
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Graphical Interpretation
Graphical interpretation involves analyzing the behavior of functions through their graphs. By plotting the functions involved in the Sandwich Theorem, one can visually assess how they behave as x approaches a specific value, such as 0. This can provide insights into the limits and continuity of the functions, enhancing understanding of their relationships.
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