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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1:57 minutes

Problem 2.4.27

Textbook Question

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (tan 2x) / x

Verified step by step guidance

Recognize that the limit involves the function tan(2x), which can be rewritten using the identity tan(2x) = sin(2x) / cos(2x). This gives us the expression lim(x→0) (sin(2x) / (x * cos(2x))).

To simplify the expression, focus on the term sin(2x) / x. Notice that this can be rewritten as (sin(2x) / 2x) * (2x / x).

Apply the limit property lim(θ→0) (sin(θ) / θ) = 1 to the term (sin(2x) / 2x). This implies that as x approaches 0, (sin(2x) / 2x) approaches 1.

Now, consider the term (2x / x), which simplifies to 2. Therefore, the expression (sin(2x) / x) simplifies to 2 * (sin(2x) / 2x).

Finally, evaluate the limit of the entire expression: lim(x→0) (2 * (sin(2x) / 2x) / cos(2x)). Since (sin(2x) / 2x) approaches 1 and cos(2x) approaches cos(0) = 1 as x approaches 0, the limit simplifies to 2 * 1 / 1.

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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 certain value. In calculus, understanding limits is crucial for analyzing how functions behave near specific points, which is foundational for defining derivatives and integrals. The limit limθ→0 sin θ / θ = 1 is a standard result used to evaluate limits involving trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They are essential tools for simplifying expressions and solving problems in calculus. For example, the identity tan x = sin x / cos x can be used to rewrite and simplify the expression tan 2x in terms of sine and cosine, aiding in the evaluation of limits.

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(x)/g(x) as x approaches a point results in an indeterminate form, the limit can be found by differentiating the numerator and denominator separately. This rule is particularly useful for evaluating limits involving complex expressions, such as limx→0 (tan 2x) / x.