Ch. 2 - Limits and Continuity

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

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.4.26

Chapter 2, Problem 2.4.26

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0− h / sin 3h

Verified step by step guidance

Recognize that the limit limθ→0 sin θ / θ = 1 is a standard limit in calculus, which can be used to evaluate limits involving trigonometric functions.

Rewrite the given limit limh→0− h / sin 3h in a form that resembles the standard limit. Notice that the expression involves sin 3h, so we need to manipulate it to use the standard limit.

Factor out the constant inside the sine function by rewriting the expression as limh→0− (1/3) * (3h / sin 3h). This step involves recognizing that multiplying and dividing by 3 will help us match the standard limit form.

Apply the standard limit limθ→0 sin θ / θ = 1 to the expression 3h / sin 3h. As h approaches 0, 3h also approaches 0, allowing us to use the standard limit.

Conclude that the limit is (1/3) * 1, since the limit of 3h / sin 3h as h approaches 0 is 1. Therefore, the original limit evaluates to 1/3.

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Key Concepts

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

Limit of a Function

The limit of a function describes the behavior of that function as its input approaches a certain value. In calculus, limits are fundamental for defining derivatives and integrals. Understanding limits helps in evaluating expressions that may be indeterminate or undefined at specific points.

Sine Function and Its Properties

The sine function, denoted as sin(θ), is a periodic function that relates the angle θ to the ratio of the opposite side to the hypotenuse in a right triangle. A key property of the sine function is that as θ approaches 0, sin(θ) approaches θ, which is crucial for evaluating limits involving sin(θ) and θ.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule simplifies the process of finding limits in complex expressions.

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Textbook Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 x sin (1/x) = 0

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Textbook Question

Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = 4 + 3x² / (x² + 1)

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 → ±∞ f(x) = 0, lim x → 2⁻ f(x) = ∞, and lim x → 2⁺ f(x) = ∞

Textbook Question

Horizontal and Vertical Asymptotes

Determine the domain and range of y = (√16―x²) / (x―2).

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Textbook Question

Limits of Rational Functions

In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.

f(x) = (x + 1)/(x² + 3)

Textbook Question

Limits with trigonometric functions

Find the limits in Exercises 43–50.