Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.5.60

Chapter 2, Problem 2.5.60

If functions f(x) and g(x) are continuous for 0 ≤ x ≤ 1, could f(x)/g(x) possibly be discontinuous at a point of [0,1]? Give reasons for your answer.

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To determine if f(x)/g(x) could be discontinuous, we need to consider the conditions under which a quotient of two continuous functions might be discontinuous.

Recall that a function is discontinuous at a point if it is not defined or if the limit does not exist or does not equal the function's value at that point.

For f(x)/g(x) to be discontinuous, g(x) must be zero at some point in the interval [0,1] because division by zero is undefined.

Even if g(x) is zero at a point, f(x)/g(x) could still be continuous if the limit of f(x)/g(x) as x approaches that point exists and equals the value of the function at that point.

Therefore, f(x)/g(x) could be discontinuous at a point in [0,1] if g(x) is zero at that point and the limit of f(x)/g(x) as x approaches that point does not exist or does not equal the function's value at that point.

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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 functions f(x) and g(x) to be continuous on the interval [0, 1], they must not have any breaks, jumps, or asymptotes within that range. This property is crucial for understanding how the behavior of these functions affects their quotient.

Quotient of Functions

The quotient of two functions, f(x)/g(x), is defined wherever g(x) is not equal to zero. If g(x) approaches zero at any point in the interval [0, 1], the quotient may become undefined or exhibit discontinuity at that point. Thus, the behavior of g(x) is critical in determining the continuity of the quotient function.

Discontinuity in Quotients

A function can be discontinuous if it involves division by a function that is zero at some point. Even if f(x) and g(x) are continuous, if g(x) equals zero at a point in [0, 1], then f(x)/g(x) will be discontinuous at that point. This highlights the importance of analyzing the denominator in the context of continuity.

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Related Practice

Textbook Question

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Prove the limit statements in Exercises 37–50.

lim x→0 x sin (1/x) = 0

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Textbook Question

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)

lim x → ±∞ f(x) = 0, lim x → 2⁻ f(x) = ∞, and lim x → 2⁺ f(x) = ∞

Textbook Question

Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

x³ − 15x + 1 = 0 (three roots)

Textbook Question

Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = 2x / (x² − 1)

Textbook Question

Finding Limits

In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.

lim (4g(x))¹/³ = 2

x →0

Textbook Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 x² sin (1/x) = 0

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