Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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2:02 minutes

Problem 2.4.36

Textbook Question

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.