Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problema 2.8a

Capitolo 2, Problema 2.8a

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

a. ƒ(x) = tan x

Guida verificata passo dopo passo

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 \( k \).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.

Video consigliato:

05:50

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.

Video consigliato:

05:34

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.

Video consigliato:

Percorso guidato

05:13

Slopes of Tangent Lines

Pratica correlata

Domanda del libro di testo

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)

365

views

Domanda del libro di testo

[Technology Exercise] Roots

Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.

a. Use the Intermediate Value Theorem to show that ƒ has a zero between ―1 and 2 .

184

views

Domanda del libro di testo

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 .

241

views

Domanda del libro di testo

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.

226

views

Domanda del libro di testo

Average Rates of Change

In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.

g(x)=x²−2x

a. [1, 3]

429

views

Domanda del libro di testo

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))