Skip to main content
Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.6.25
Chapter 2, Problem 2.6.25

Determine the interval(s) on which the following functions are continuous. 
p(x)=4x^5−3x^2+1

Verified step by step guidance
1
p>Step 1: Recognize that the function \( p(x) = 4x^5 - 3x^2 + 1 \) is a polynomial function.
p>Step 2: Recall that polynomial functions are continuous everywhere on the real number line.
p>Step 3: Conclude that the function \( p(x) \) is continuous for all real numbers.
p>Step 4: Express the interval of continuity for \( p(x) \) as \((-\infty, \infty)\).
p>Step 5: Verify that there are no restrictions or discontinuities in the function \( p(x) \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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 means there are no breaks, jumps, or asymptotes in the function's graph.
Recommended video:
05:34
Intro to Continuity

Polynomial Functions

Polynomial functions, like p(x) = 4x^5 - 3x^2 + 1, are continuous everywhere on the real number line. This is because they are composed of terms that are powers of x with real coefficients, which do not introduce any discontinuities. Understanding the nature of polynomial functions is crucial for determining their continuity.
Recommended video:
6:04
Introduction to Polynomial Functions

Intervals of Continuity

Intervals of continuity refer to the ranges of x-values over which a function remains continuous. For polynomial functions, the interval of continuity is typically all real numbers, denoted as (-∞, ∞). Identifying these intervals involves analyzing the function's behavior and ensuring it meets the criteria for continuity across the specified range.
Recommended video:
03:38
Intro to Continuity Example 1
Related Practice
Textbook Question

Evaluate each limit. 


limx→πcos2(x)+3cos(x)+2cos(x)+1{\(\displaystyle\]\lim\)_{x\(\to\[\pi\)}\(\frac{\cos^2\left(x\right)+3\cos\left(x\right)+2}{\cos^{}\]\left\)(x\(\right\))+1}}

316
views
Textbook Question

Suppose limx→a f(x)=L{\(\displaystyle\]\lim\)_{x\(\to\) a}}\(\text{ }\)f\(\left\)(x\(\right\))=L. Prove that limx→a (c(f(x))=cL{\(\displaystyle\]\lim\)_{x\(\to\) a}}\(\text{ (}\)c\(\left\)(f\(\left\)(x\(\right\))\(\right\))=cL, where cc is a constant.

368
views
Textbook Question

Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. <IMAGE>

336
views
Textbook Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.


lim p→2 3p / √4p + 1 − 1

283
views
Textbook Question

Suppose |f(x) − 5|<0.1 whenever 0<x<5. Find all values of δ>0 such that |f(x) − 5|<0.1 whenever 0<|x−2|<δ.

361
views
Textbook Question

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 


f(z)=(z−1)^3/4

257
views