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

3:11 minutes

Problem 2.8a

Textbook Question

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

a. ƒ(x) = tan x

Verified step by step guidance

Identify the function: The function given is \( f(x) = \tan x \). The tangent function is defined as \( \tan x = \frac{\sin x}{\cos x} \).

Determine where the function is undefined: The function \( \tan x \) is undefined where \( \cos x = 0 \) because division by zero is undefined.

Find the values of \( x \) where \( \cos x = 0 \): The cosine function is zero at odd multiples of \( \frac{\pi}{2} \), i.e., \( x = \frac{\pi}{2} + k\pi \) where \( k \) is an integer.

Identify intervals of continuity: The function \( \tan x \) is continuous on intervals where it is defined, which are the open intervals between the points where \( \cos x = 0 \).

Conclude the intervals of continuity: Therefore, \( f(x) = \tan x \) is continuous on intervals of the form \( (k\pi - \frac{\pi}{2}, k\pi + \frac{\pi}{2}) \) for each integer \( k \).

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.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. Understanding limits is crucial for analyzing the behavior of functions, especially at points where they may not be explicitly defined. For example, the limit of tan(x) as x approaches π/2 is undefined, indicating a vertical asymptote.

Continuity

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. Discontinuities can occur due to vertical asymptotes, jumps, or holes in the graph, which are essential to identify when determining intervals of continuity.

The Tangent Function

The tangent function, defined as tan(x) = sin(x)/cos(x), is periodic and has vertical asymptotes where the cosine function equals zero, specifically at odd multiples of π/2. This periodicity and the locations of its discontinuities are critical for determining the intervals of continuity for tan(x). Thus, tan(x) is continuous on intervals that do not include these asymptotes.

Watch next

Master Intro to Continuity with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

[Technology Exercise] In Exercises 33–36, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension, can it be extended to be continuous from the right or left? If so, what do you think the extended function’s value should be?

g(θ) = 5 cos θ / (4θ ― 2π) , a = π/2