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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 2.5.24
Chapter 2, Problem 2.5.24

At what points are the functions in Exercises 13–30 continuous?
y = √(x⁴ +1)/(1 + sin² x)

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1
Identify the components of the function \( y = \frac{\sqrt{x^4 + 1}}{1 + \sin^2 x} \). The numerator is \( \sqrt{x^4 + 1} \) and the denominator is \( 1 + \sin^2 x \).
Determine the domain of the numerator \( \sqrt{x^4 + 1} \). Since \( x^4 + 1 \) is always positive for all real numbers, \( \sqrt{x^4 + 1} \) is defined for all \( x \in \mathbb{R} \).
Examine the denominator \( 1 + \sin^2 x \). The function \( \sin^2 x \) is defined for all real numbers and always non-negative, making \( 1 + \sin^2 x \) always positive. Therefore, the denominator is never zero for any real \( x \).
Since both the numerator and the denominator are defined for all real numbers and the denominator is never zero, the function \( y = \frac{\sqrt{x^4 + 1}}{1 + \sin^2 x} \) is continuous for all \( x \in \mathbb{R} \).
Conclude that the function is continuous everywhere on the real line, as there are no points where the function is undefined or where the denominator is zero.

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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. For a function to be continuous over an interval, it must be continuous at every point within that interval.
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Intro to Continuity

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 given function, we need to identify any values of x that might cause the denominator to be zero or lead to undefined expressions, as these points will affect continuity.
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Finding the Domain and Range of a Graph

Behavior of Square Roots and Trigonometric Functions

The square root function is defined for non-negative values, meaning the expression inside the square root must be greater than or equal to zero. Additionally, the sine function oscillates between -1 and 1, affecting the overall behavior of the function. Understanding these behaviors helps determine where the function remains continuous.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

Limits of quotients


Find the limits in Exercises 23–42.


limu→1 (u⁴ − 1)/(u³ − 1)

Textbook Question

Formal Definitions of One-Sided Limits


Greatest integer function Find (a) limx→400+ ⌊x⌋ and (b) limx→400− ⌊x⌋; then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about limx→400 ⌊x⌋? Give reasons for your answer.

Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limx→0 (x² − x + sin x) / 2x

Textbook Question

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx→−3 (x² − 9) / (x + 3) = −6

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 − 3x − 1 = 0

Textbook Question

Suppose that a function f(x) is defined for all x in [-1,1]. Can anything be said about the existence of limx→0 f(x)? Give reasons for your answer.