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

Previous problemNext problem

1:16 minutes

Problem 2.8c

Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?

c. h(x) = cos x / x―π

Verified step by step guidance

Identify the type of function: The function h(x) = cos(x) / (x - π) is a rational function, which is generally continuous everywhere except where the denominator is zero.

Determine where the denominator is zero: Set the denominator equal to zero to find the points of discontinuity. Solve the equation x - π = 0, which gives x = π.

Analyze the continuity: Since the function is a rational function, it is continuous on its domain, which is all real numbers except where the denominator is zero.

Express the intervals of continuity: The function is continuous on the intervals (-∞, π) and (π, ∞), as these intervals do not include the point where the denominator is zero.

Conclude the analysis: Therefore, the function h(x) = cos(x) / (x - π) is continuous on the intervals (-∞, π) and (π, ∞).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. This concept is crucial for determining where a function does not have breaks, jumps, or asymptotes.

Limits

Limits describe the behavior of a function as it approaches a specific point from either side. Understanding limits is essential for analyzing continuity, especially at points where the function may not be explicitly defined, such as points of division by zero. Evaluating limits helps identify potential discontinuities in the function.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals. When analyzing continuity for such functions, it is important to check the limits and values at the boundaries of these intervals. This concept is relevant for determining the overall continuity of functions that may behave differently in separate segments.