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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 2.8b
Chapter 2, Problem 2.8b

Limits and Continuity


On what intervals are the following functions continuous?


b. g(x) = csc x

Verified step by step guidance
1
Identify the function: The function given is \( g(x) = \csc x \), which is the cosecant function. Recall that \( \csc x = \frac{1}{\sin x} \).
Determine where the function is undefined: Since \( \csc x \) is the reciprocal of \( \sin x \), it is undefined wherever \( \sin x = 0 \).
Find the zeros of \( \sin x \): The sine function is zero at integer multiples of \( \pi \), i.e., \( x = n\pi \) where \( n \) is an integer.
Identify intervals of continuity: The function \( g(x) = \csc x \) is continuous on intervals where \( \sin x \neq 0 \). These intervals are between the points where \( \sin x = 0 \), i.e., between \( n\pi \) and \( (n+1)\pi \) for any integer \( n \).
Conclude the intervals: Therefore, \( g(x) = \csc x \) is continuous on the intervals \( (n\pi, (n+1)\pi) \) for all integers \( n \).

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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 functions do not have breaks, jumps, or asymptotes.
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Intro to Continuity

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). It is important to note that csc(x) is undefined wherever sin(x) = 0, which occurs at integer multiples of π. Understanding the behavior of the cosecant function helps identify the intervals of continuity.
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Graphs of Secant and Cosecant Functions

Intervals of Continuity

Intervals of continuity refer to the ranges of x-values where a function is continuous. For the function g(x) = csc(x), we need to exclude points where the function is undefined, specifically at x = nπ (where n is an integer). By identifying these points, we can determine the intervals where g(x) is continuous, which are the open intervals between these undefined points.
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Related Practice
Textbook Question

Theory and Examples


If limx→−2 f(x) / x² = 1, find


b. limx→−2 f(x) / x

Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?


b. g(x) = x³/⁴

Textbook Question

Estimating Limits


[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.


Let f(x) = (x² - 9) / (x + 3)


b. Support your conclusions in part (a) by graphing f near c = -3 and using Zoom and Trace to estimate y-values on the graph as x → −3.

Textbook Question

Estimating Limits


[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.


Let G(x)=(x + 6)/(x² + 4x − 12)


b. Support your conclusions in part (a) by graphing G and using Zoom and Trace to estimate y-values on the graph as x→−6.

Textbook Question

Finding Limits Graphically


Which of the following statements about the function y = f(x) graphed here are true, and which are false?



b. limx→2 f(x) does not exist

Textbook Question

Estimating Limits


[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.


Let g(θ) = (sinθ) / θ.


b. Support your conclusion in part (a) by graphing g near θ₀ = 0.