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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.R.78
Chapter 2, Problem 2.R.78

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

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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.

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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.
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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.
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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.
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Related Practice
Textbook Question

The height above the ground of a stone thrown upwards is given by s(t), where t is measured in seconds. After 1 second, the height of the stone is 48 feet above the ground, and after 1.5 seconds, the height of the stone is 60 feet above the ground. Evaluate s(1) and s(1.5), and then find the average velocity of the stone over the time interval [1, 1.5].

Textbook Question

b. Estimate a solution to the equation in the given interval using a root finder.


x=cos x; (0,π/2)

Textbook Question

Determine the following limits.

lim x→∞ (3 tan-1 x + 2)

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Textbook Question

Determine the following limits.

lim x→∞ (2x − 3) / (4x + 10)

Textbook Question

Use the graph of ff in the figure to determine the values of xx in the interval (−3,5)\(\left\)(-3,5\(\right\)) at which f fails to be continuous. Justify your answers using the continuity checklist.

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Textbook Question

Let g(x)={5x−2if x<1aif x=1ax2+bxif x>1g\(\left\)(x\(\right\))=\(\begin{cases}\)5x-2 & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ ax^2+bx & \(\text{if }\)x>1\(\end{cases}\).


Determine values of the constants aa and bb , if possible, for which gg is continuous at x=1x=1.