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

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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1:11 minutes

Problem 2.2.43

Textbook Question

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→0 (2sin x − 1)

Verified step by step guidance

Identify the limit expression: \( \lim_{x \to 0} (2\sin x - 1) \).

Recall the basic limit property for sine: \( \lim_{x \to 0} \sin x = 0 \).

Substitute the limit of \( \sin x \) into the expression: \( 2(\lim_{x \to 0} \sin x) - 1 \).

Simplify the expression using the known limit: \( 2(0) - 1 \).

Conclude the limit evaluation by simplifying the expression to find the result.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, we are interested in the limit of the function as x approaches 0, which helps determine the behavior of the function near that point.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate angles to ratios of sides in right triangles. They are periodic functions and play a crucial role in calculus, especially when evaluating limits involving angles. The sine function, in particular, has specific limit properties that are useful when x approaches 0, such as sin(x) approaching x.

Sine Limit Property

The sine limit property states that as x approaches 0, the limit of sin(x)/x equals 1. This property is pivotal when evaluating limits involving sine functions, as it allows simplification of expressions. In the given limit, understanding this property will help in determining the limit of the expression 2sin(x) as x approaches 0.