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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.5.13
Chapter 2, Problem 2.5.13

At what points are the functions in Exercises 13–30 continuous?
y = 1/(x – 2) – 3x

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Identify the function given: \( y = \frac{1}{x - 2} - 3x \). This function is composed of two parts: a rational function \( \frac{1}{x - 2} \) and a linear function \( -3x \).
Determine the points of discontinuity for the rational function \( \frac{1}{x - 2} \). A rational function is discontinuous where its denominator is zero. Set the denominator equal to zero: \( x - 2 = 0 \).
Solve the equation \( x - 2 = 0 \) to find the point of discontinuity. This gives \( x = 2 \). Therefore, the function is discontinuous at \( x = 2 \).
Consider the linear function \( -3x \). Linear functions are continuous everywhere on the real number line, so there are no additional points of discontinuity from this part of the function.
Conclude that the function \( y = \frac{1}{x - 2} - 3x \) is continuous for all real numbers except \( x = 2 \).

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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 it is defined at that point, the limit exists at that point, and the limit equals the function's value at that point. For rational functions, continuity is typically disrupted by points where the denominator equals zero, leading to undefined values.
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Intro to Continuity

Identifying Discontinuities

To determine where a function is continuous, one must identify points of discontinuity, which occur when the denominator of a rational function is zero. For the given function, setting the denominator (x - 2) to zero reveals potential discontinuities, which must be analyzed further.
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Intro to Continuity Example 1

Limits

Limits are fundamental in calculus for understanding the behavior of functions as they approach specific points. Evaluating limits helps determine if a function approaches a finite value or diverges, which is crucial for assessing continuity at points where the function may be undefined.
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One-Sided Limits
Related Practice
Textbook Question

Infinite Limits


Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.


lim x→0⁺ 1 / 3x

Textbook Question

Calculating Limits


Find the limits in Exercises 11–22.


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Using the Formal Definition


Prove the limit statements in Exercises 37–50.


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Textbook Question

At what points are the functions in Exercises 13–30 continuous?


g(x) = { (x² − x – 6)/(x – 3), x ≠ 3

5, x = 3


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Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)


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Limits and Infinity


Find the limits in Exercises 37–46.


x²/³ + x⁻¹

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