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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 2.8a
Chapter 2, Problem 2.8a

Limits and Continuity
On what intervals are the following functions continuous?


a. ƒ(x) = tan x

Verified step by step guidance
1
Identify the function: The function given is \( f(x) = \tan x \). The tangent function is defined as \( \tan x = \frac{\sin x}{\cos x} \).
Determine where the function is undefined: The function \( \tan x \) is undefined where \( \cos x = 0 \) because division by zero is undefined.
Find the values of \( x \) where \( \cos x = 0 \): The cosine function is zero at odd multiples of \( \frac{\pi}{2} \), i.e., \( x = \frac{\pi}{2} + k\pi \) where \( k \) is an integer.
Identify intervals of continuity: The function \( \tan x \) is continuous on intervals where it is defined, which are the open intervals between the points where \( \cos x = 0 \).
Conclude the intervals of continuity: Therefore, \( f(x) = \tan x \) is continuous on intervals of the form \( (k\pi - \frac{\pi}{2}, k\pi + \frac{\pi}{2}) \) for each integer \( k \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. Understanding limits is crucial for analyzing the behavior of functions, especially at points where they may not be explicitly defined. For example, the limit of tan(x) as x approaches π/2 is undefined, indicating a vertical asymptote.
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One-Sided Limits

Continuity

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. Discontinuities can occur due to vertical asymptotes, jumps, or holes in the graph, which are essential to identify when determining intervals of continuity.
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Intro to Continuity

The Tangent Function

The tangent function, defined as tan(x) = sin(x)/cos(x), is periodic and has vertical asymptotes where the cosine function equals zero, specifically at odd multiples of π/2. This periodicity and the locations of its discontinuities are critical for determining the intervals of continuity for tan(x). Thus, tan(x) is continuous on intervals that do not include these asymptotes.
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Related Practice
Textbook Question

Limits of Rational Functions


In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.


f(x) = (x + 1)/(x² + 3)

Textbook Question

Limits and Continuity

Repeat the instructions of Exercise 1 for


1 , x ≤ ―1

1/x , 0 < |x| < 1

ƒ(x) = { 0, x = 1 ,

1 , x > 1 .

Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?


a. ƒ(x) = x¹/³

Textbook Question

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.


lim ( 1 / x²/³ + 2 / (x − 1)²/³ ) as


a. x → 0⁺

Textbook Question

Limits and Continuity


Graph the function


1 , x ≤ ―1

―x , ―1 < x < 0

ƒ(x) = { 1 , x = 0 ,

―x , 0 < x < 1

1 , x ≥ 1


Then discuss, in detail, limits, one-sided limits, continuity, and one-sided continuity of ƒ at x = ―1 , 0 , and 1. Are any of the discontinuities removable? Explain.

Textbook Question

Suppose that limx→−2 p(x) = 4, limx→−2 r(x) = 0, and limx→−2 s(x) = −3. Find


a. limx→−2 (p(x) + r(x) + s(x))