Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.43

Chapter 2, Problem 2.43

Limits and Infinity

Find the limits in Exercises 37–46.

sin x
lim ------------- ( If you have a grapher, try graphing
x→∞ |x| the function for ―5 ≤ x ≤ 5 ) .

Verified step by step guidance

Identify the function for which you need to find the limit: \( \frac{\sin x}{|x|} \).

Understand that as \( x \to \infty \), the absolute value \( |x| \) behaves like \( x \) because \( x \) is positive.

Recognize that the sine function, \( \sin x \), oscillates between -1 and 1 for all real numbers \( x \).

Consider the behavior of the fraction \( \frac{\sin x}{x} \) as \( x \to \infty \). Since \( \sin x \) is bounded and \( x \) grows without bound, the fraction approaches zero.

Conclude that the limit of \( \frac{\sin x}{|x|} \) as \( x \to \infty \) is 0, because the numerator is bounded while the denominator increases indefinitely.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They help in understanding the behavior of functions at specific points, including points of discontinuity or infinity. In this context, evaluating the limit as x approaches infinity allows us to analyze the long-term behavior of the function sin(x)/|x|.

Behavior of Functions at Infinity

When analyzing limits as x approaches infinity, we assess how a function behaves as its input grows without bound. This often involves determining whether the function approaches a finite value, diverges to infinity, or oscillates. For the function sin(x)/|x|, understanding its behavior as x becomes very large is crucial for finding the limit.

Trigonometric Functions

Trigonometric functions, such as sine, exhibit periodic behavior, oscillating between fixed values. The function sin(x) oscillates between -1 and 1, which is important when considering its limit in conjunction with |x|. This periodic nature influences the overall limit of the function sin(x)/|x| as x approaches infinity, as the oscillation will be divided by an increasingly large denominator.

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Related Practice

Textbook Question

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

lim x → −5 (1 / (x + 5)²) = ∞

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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))

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Textbook Question

Limits of quotients

Find the limits in Exercises 23–42.

limx→1 (x −1) / (√(x + 3) − 2)

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Textbook Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.