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

Continuity

Calculus

1. Limits and Continuity

Continuity

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1:38 minutes

Problem 2.8d

Textbook Question

Limits and Continuity
On what intervals are the following functions continuous?

d. k(x) = sin x / x

Verified step by step guidance

First, recall the definition of continuity: a function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point.

Consider the function k(x) = sin(x) / x. This function is defined for all x except x = 0, because division by zero is undefined.

To determine the intervals of continuity, we need to check where the function is defined and where it is continuous. Since k(x) is undefined at x = 0, it cannot be continuous there.

For x ≠ 0, the function is continuous because both sin(x) and x are continuous functions, and the quotient of two continuous functions is continuous wherever the denominator is non-zero.

Therefore, the function k(x) = sin(x) / x is continuous on the intervals (-∞, 0) and (0, ∞).

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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 crucial for understanding continuity, as a function is continuous at a point if the limit exists and equals the function's value at that point. For the function k(x) = sin x / x, evaluating the limit as x approaches 0 is essential to determine its continuity.

Continuity

A function is continuous at a point if three conditions are met: the function is defined at that point, the limit exists at that point, and the limit equals the function's value. For k(x) = sin x / x, we need to check its behavior at x = 0 and other points to determine where it is continuous. Understanding the definition of continuity helps in identifying intervals of continuity for various functions.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals. For k(x) = sin x / x, it is important to recognize that while the function is not defined at x = 0, it can be extended to be continuous by defining k(0) = 1, which is the limit of k(x) as x approaches 0. This concept is vital for analyzing functions that may have discontinuities at specific points.