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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.2.66
Chapter 2, Problem 2.2.66

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 guidance
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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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Fundamental Theorem of Calculus Part 1

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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Limits of Rational Functions: Denominator = 0

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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Related Practice
Textbook Question

In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?

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Textbook Question

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)


lim x → ±∞ k(x) = 1, lim x → 1⁻ k(x) = ∞, and lim x → 1⁺ k(x) = −∞

Textbook Question

Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find


b. limx→c 2f(x)g(x)

Textbook Question

Use formal definitions to prove the limit statements in Exercises 93–96.


lim x → −5 (1 / (x + 5)²) = ∞

Textbook Question

Finding Limits


In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)


g(x) = 1/(2 + (1/x))

Textbook Question

Limits of quotients


Find the limits in Exercises 23–42.


limx→1 (x −1) / (√(x + 3) − 2)