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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1:31 minutes

Problem 21

Textbook Question

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.

lim x →π sin (x/2 + sin x)

Verified step by step guidance

Step 1: Understand the problem. We need to find the limit of the function sin(x/2 + sin(x)) as x approaches π.

Step 2: Substitute x = π into the expression inside the sine function. This gives us π/2 + sin(π). Since sin(π) = 0, the expression simplifies to π/2.

Step 3: Evaluate the sine function at π/2. We know that sin(π/2) = 1.

Step 4: Since the function sin(x/2 + sin(x)) is continuous around x = π, the limit as x approaches π is simply the value of the function at x = π.

Step 5: Conclude that the limit of sin(x/2 + sin(x)) as x approaches π is 1, based on the continuity and substitution.

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Key Concepts

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

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for defining derivatives and integrals. Limits can exist or not exist, and determining their existence often involves evaluating the function at points close to the limit.

Continuity

Continuity refers to a property of functions where they do not have any abrupt changes, jumps, or holes at a given point. A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. Understanding continuity is essential for evaluating limits, as discontinuities can lead to limits that do not exist.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In the context of limits, these functions can exhibit specific behaviors as their arguments approach certain values, which can affect the limit's existence. Familiarity with the properties and values of trigonometric functions is crucial for evaluating limits involving them.