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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1:38 minutes

Problem 2.4.39

Textbook Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 θcos θ

Verified step by step guidance

Recognize that the limit involves a trigonometric function and a variable approaching zero. The expression given is \( \lim_{\theta \to 0} \theta \cos \theta \).

Understand that \( \cos \theta \) is a continuous function, and as \( \theta \to 0 \), \( \cos \theta \to \cos(0) = 1 \).

Rewrite the limit expression by separating the terms: \( \lim_{\theta \to 0} \theta \cos \theta = \lim_{\theta \to 0} \theta \cdot \lim_{\theta \to 0} \cos \theta \).

Substitute the known limit of \( \cos \theta \) as \( \theta \to 0 \), which is 1, into the expression: \( \lim_{\theta \to 0} \theta \cdot 1 \).

Evaluate the limit of \( \theta \) as \( \theta \to 0 \), which is 0. Therefore, the entire expression evaluates to \( 0 \cdot 1 = 0 \).

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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 value that a function approaches as the input approaches a certain point. In calculus, understanding limits is crucial for analyzing the behavior of functions near specific points, especially when direct substitution may lead to indeterminate forms.

Trigonometric Limits

Trigonometric limits often involve functions like sine and cosine, particularly as the angle approaches zero. A key result is that limθ→0 sin θ / θ = 1, which is foundational for evaluating limits involving trigonometric functions and is frequently used in calculus to simplify expressions.

Continuity and Differentiability

A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. Differentiability, which requires continuity, is essential for understanding how functions behave and change, particularly when evaluating limits that involve products of functions, such as θcos θ.