Ch. 2 - Limits and Continuity

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

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.4.36

Chapter 2, Problem 2.4.36

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0 sin(sin h) / sin h

Verified step by step guidance

First, recognize that the problem involves finding the limit of the expression lim(h→0) sin(sin(h)) / sin(h). This is a limit problem that can be approached using the known limit lim(θ→0) sin(θ) / θ = 1.

To solve this, consider the substitution u = sin(h). As h approaches 0, u also approaches 0 because sin(h) is continuous and sin(0) = 0.

Rewrite the original limit in terms of u: lim(u→0) sin(u) / u. This transformation is valid because as h approaches 0, u approaches 0.

Now, apply the known limit property: lim(θ→0) sin(θ) / θ = 1. Since u is approaching 0, this property can be directly applied to the expression lim(u→0) sin(u) / u.

Conclude that the limit of the original expression lim(h→0) sin(sin(h)) / sin(h) is equal to 1, based on the substitution and the known limit property.

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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 the function as the input approaches a particular value. In calculus, understanding limits is crucial for analyzing the continuity and differentiability of functions. For the given problem, evaluating the limit as h approaches 0 is essential to determine the behavior of the function sin(sin h) / sin h.

Squeeze Theorem

The Squeeze Theorem is a method used to find the limit of a function by 'squeezing' it between two other functions whose limits are known and equal. This theorem is particularly useful when direct substitution in a limit results in an indeterminate form. In this problem, the Squeeze Theorem can help confirm the limit of sin(sin h) / sin h as h approaches 0.

Limit of sin(θ)/θ as θ approaches 0

The limit limθ→0 sin(θ)/θ = 1 is a fundamental result in calculus, often used to evaluate limits involving trigonometric functions. This limit is derived from the unit circle and the small-angle approximation. In the given problem, this concept is directly applied to simplify and solve the limit of sin(sin h) / sin h as h approaches 0.

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Using limθ→0 sin θ / θ = 1

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Textbook Question

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The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

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Using limθ→0 sin θ / θ = 1Find the limits in Exercises 23–46.limh→0 sin(sin h) / sin h

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Key Concepts

Limit of a Function

Squeeze Theorem

Limit of sin(θ)/θ as θ approaches 0

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0 sin(sin h) / sin h