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

2:50 minutes

Problem 2.R.78

Textbook Question

Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.
$g (x) = \cos (e^{x})$

Verified step by step guidance

Step 1: Identify the components of the function g(x) = cos(e^x). The function is composed of an exponential function e^x and a trigonometric function cos(x).

Step 2: Determine the continuity of the inner function e^x. The exponential function e^x is continuous for all real numbers x.

Step 3: Determine the continuity of the outer function cos(x). The cosine function is continuous for all real numbers.

Step 4: Use the composition of continuous functions theorem. Since both e^x and cos(x) are continuous for all real numbers, their composition g(x) = cos(e^x) is also continuous for all real numbers.

Step 5: Conclude that g(x) = cos(e^x) is continuous on the interval (-∞, ∞). There are no finite endpoints to consider for right- or left-continuity.

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.

Endpoints and Continuity

When analyzing the continuity of functions on closed intervals, special attention must be given to the endpoints. A function can be left-continuous at the left endpoint and right-continuous at the right endpoint. Understanding how a function behaves at these endpoints is essential for accurately describing its continuity over the entire interval.

Behavior of Composite Functions

The function g(x) = cos(e^x) is a composite function, where the continuity of g depends on the continuity of both the cosine function and the exponential function. Since both functions are continuous everywhere, g(x) is also continuous for all real numbers. Recognizing how the continuity of inner and outer functions affects the overall function is vital in calculus.

Watch next

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

Start learning

Related Videos

Related Practice

Textbook Question

Determine the interval(s) on which the following functions are continuous.

f(t)=t+2 / t^2−4

Textbook Question

Let $g\left(x\right)=\begin{cases}x^2+x & \text{if }x<1\\ a & \text{if }x=1\\ 3x+5 & \text{if }x>1\end{cases}$

b. Determine the value of $a$ for which $g$ is continuous from the right at $1$ .

Textbook Question

Let $g\left(x\right)=\begin{cases}x^2+x & \text{if }x<1\\ a & \text{if }x=1\\ 3x+5 & \text{if }x>1\end{cases}$

a. Determine the value of a for which $g$ is continuous from the left at $1$ .

Textbook Question

Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.

$h (x) = \frac{2 x}{x^{3} - 25 x}$

Textbook Question

Suppose the rental cost for a snowboard is $25 for the first day (or any part of the first day) plus $15 for each additional day (or any part of a day).

e. For what values of t is f continuous? Explain.

Textbook Question

Let $g\left(x\right)=\begin{cases}5x-2 & \text{if }x<1\\ a & \text{if }x=1\\ ax^2+bx & \text{if }x>1\end{cases}$ .