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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 2.5.25
Chapter 2, Problem 2.5.25

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


y = √(2x + 3)

Verified step by step guidance
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Step 1: Understand the concept of continuity. A function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point.
Step 2: Identify the domain of the function y = √(2x + 3). The expression inside the square root, 2x + 3, must be greater than or equal to zero for the function to be defined.
Step 3: Solve the inequality 2x + 3 ≥ 0 to find the values of x for which the function is defined. This will give you the domain of the function.
Step 4: Since the square root function is continuous wherever it is defined, the function y = √(2x + 3) will be continuous for all x within the domain found in Step 3.
Step 5: Conclude that the function is continuous at all points within the domain determined by the inequality 2x + 3 ≥ 0.

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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. This means there are no breaks, jumps, or holes in the graph of the function at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval.
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Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function y = √(2x + 3), the expression under the square root must be non-negative, which imposes restrictions on the values of x. Identifying the domain is crucial for determining where the function is continuous.
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Square Root Function Properties

The square root function, such as y = √(x), is defined only for non-negative values of x. This means that for the function y = √(2x + 3), we need to ensure that the expression 2x + 3 is greater than or equal to zero. Understanding these properties helps in finding the intervals where the function is continuous.
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Related Practice
Textbook Question

Limits and Infinity


Find the limits in Exercises 37–46.


2x² + 3

lim -------------

x→⁻∞ 5x² + 7

Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limt→0 sin(1 − cos t) / (1 − cos t)

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Textbook Question

Using the Sandwich Theorem


a. It can be shown that the inequalities 1 − x²/ 6 < (x sin x) / (2−2cos x) < 1 hold for all values of x close to zero (except for x = 0). What, if anything, does this tell you about limx→0 (x sin x) / (2 − 2cos x)?


Give reasons for your answer.


[Technology Exercise] b. Graph y = 1 − (x²/6), y=(x sinx)/(2 − 2cos x), and y = 1 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.

Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limh→0 sin(sin h) / sin h

Textbook Question

Limits and Infinity


Find the limits in Exercises 37–46.


x⁴ + x³

lim -----------------

x→∞ 12x³ + 128

Textbook Question

Limits as x → ∞ or x → −∞


The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.


lim x→∞ (2√x + x⁻¹) / (3x − 7)