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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

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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 \( p(x) \).

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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 means there are no breaks, jumps, or asymptotes in the function's graph.
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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.
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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.
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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}}

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.

Textbook Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {√x if x<4

3 if x=4; a=4

x+1 if x>4

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Textbook Question

Determine the following limits.


lim x→π/2 1/√sin x − 1 / x + π/2

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.

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