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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2:43 minutes

Problem 2.4.37

Textbook Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 sin θ / sin 2θ

Verified step by step guidance

Recognize that the limit involves the expression \( \frac{\sin \theta}{\sin 2\theta} \) as \( \theta \to 0 \). We need to manipulate this expression to use the known limit \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 \).

Rewrite \( \sin 2\theta \) using the double angle identity: \( \sin 2\theta = 2\sin \theta \cos \theta \). This gives us \( \frac{\sin \theta}{\sin 2\theta} = \frac{\sin \theta}{2\sin \theta \cos \theta} = \frac{1}{2\cos \theta} \).

Now, consider the limit \( \lim_{\theta \to 0} \frac{1}{2\cos \theta} \). Since \( \cos \theta \to 1 \) as \( \theta \to 0 \), substitute this into the expression.

Evaluate the limit: \( \lim_{\theta \to 0} \frac{1}{2\cos \theta} = \frac{1}{2 \cdot 1} \).

Conclude that the limit is \( \frac{1}{2} \).

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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, limits are fundamental for understanding continuity, derivatives, and integrals. The notation limθ→0 indicates that we are examining the behavior of the function as θ approaches 0.

Sine Function Behavior

The sine function, sin(θ), is a periodic function that oscillates between -1 and 1. As θ approaches 0, sin(θ) behaves similarly to its argument, meaning sin(θ) approaches θ. This property is crucial for evaluating limits involving sine, particularly in the context of small angles.

L'Hôpital's Rule

L'Hôpital's Rule is a method for finding limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(θ)/g(θ) results in an indeterminate form, the limit can be evaluated by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful when direct substitution does not yield a clear result.

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Textbook Question

Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.

f(x) = x², x = 1