/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Show that \(\lim _{x \rightarrow... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 7

Show that \(\lim _{x \rightarrow c} x^{3}=c^{3}\) for any \(c \in \mathbb{R}\).

Short Answer

Expert verified
It has been shown that \(\lim _{x \rightarrow c} x^{3} = c^{3}\) for any \(c \in \mathbb{R}\). This conclusion is arrived at by understanding the given limit, substituting \(x\) for \(c\), applying limit properties for continuous functions, and confirming the result.

Step by step solution

01

Understand that limit

Understand the expression \(\lim _{x \rightarrow c} x^{3}=c^{3}\). What this means is that as \(x\) approaches the value of \(c\), the value of \(x^{3}\) approaches \(c^{3}\).
02

Substitute the limit

If you have \(\lim _{x \rightarrow c} x^{3}\), you can substitute \(x\) with \(c\) on the right side of the limit expression. The cube of any real number \(c\) will always be \(c^{3}\).
03

Apply limits property

The limit operation can be passed inside the function expression because the function \(x^{3}\) is continuous on its domain, which is \(\mathbb{R}\). Therefore, \(\lim _{x \rightarrow c} x^{3}= (\lim _{x \rightarrow c} x)^{3}\).
04

Solve limit expression

Now, the limit expression on the right \(\lim _{x \rightarrow c} x\) can be solved directly as \(c\). So, you have \((\lim _{x \rightarrow c} x)^{3} = c^{3}\). Comparing this with step 2 validates the original given limit property.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuity
Continuity in mathematics, particularly calculus, is a fundamental concept that describes how a function behaves. A function is said to be continuous at a point if the function's value at that point equals the limit of the function as it approaches that point. In simpler words, if you can draw the function's graph without lifting your pencil, then the function is continuous.
This idea is crucial because it ensures the function behaves in a predictable manner. For instance, in our exercise, the function \(x^3\) is continuous for all real numbers. It means for any real number \(c\), as \(x\) gets closer to \(c\), \(x^3\) gets closer to \(c^3\) without any sudden jumps or breaks.
The continuity of \(x^3\) is directly related to its polynomial form. Most polynomial functions are continuous across their entire domain, which means we can apply limit laws directly without adjustments.
Real numbers
Real numbers are the backbone of many mathematical concepts, including limits and continuity. They include all the numbers you can think of except imaginary numbers. Real numbers consist of rational numbers (like 1/2, 2, or -1) and irrational numbers (like \(\sqrt{2}\) or \(\pi\)). This inclusivity makes them particularly useful in calculus.
In our exercise, \(c\) is any real number, implying that the property \(\lim_{x \to c} x^3 = c^3\) holds for both rational and irrational numbers. It doesn't matter if \(c\) is 2 or \(\sqrt{2}\); the property remains valid.
Understanding real numbers is essential when working with limits, as they define the potential values both \(x\) and \(c\) can take. This ensures that limits remain applicable across a continuous and predictable range, which is why they are so fundamental in calculus.
Limit laws
Limit laws are like the rules of a game that dictate how you can manage limits in calculus. They provide a set of rules to simplify the process of finding limits, ensuring calculations are consistent and reliable.
  • Sum Law: The limit of a sum is the sum of the limits.
  • Product Law: The limit of a product is the product of the limits.
  • Power Law: The limit of a power function is the limit raised to that power.
In our exercise, we see the power law in action. It allows us to state that \(\lim_{x \to c} x^3\) can be broken down into \((\lim_{x \to c} x)^3\), thanks to the continuity of the function \(x^3\).
By applying these laws, we can find limits more intuitively and confirm that our results hold true across different scenarios in calculus.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose that \(f\) and \(g\) have limits in \(\mathbb{R}\) as \(x \rightarrow \infty\) and that \(f(x) \leq g(x)\) for all \(x \in(a, \infty)\). Prove that \(\lim _{x \rightarrow \infty} f \leq \lim _{x \rightarrow \infty} g\).

Suppose that \(\lim _{x \rightarrow c} f(x)=L\) where \(L>0\), and that \(\lim _{x \rightarrow c} g(x)=\infty\). Show that \(\lim _{x \rightarrow c} f(x) g(x)=\infty\). If \(L=0\), show by example that this conclusion may fail.

Let \(f, g\) be defined on \(A \subseteq \mathbb{R}\) to \(\mathbb{R}\), and let \(c\) be a cluster point of \(A\). Suppose that \(f\) is bounded on a neighborhood of \(c\) and that \(\lim _{x \rightarrow c} g=0\). Prove that \(\lim _{x \rightarrow c} f g=0\).

Suppose the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) has limit \(L\) at 0, and let \(a>0\). If \(g: \mathbb{R} \rightarrow \mathbb{R}\) is defined by \(g(x):=f(a x)\) for \(x \in \mathbb{R}\), show that \(\lim _{x \rightarrow 0} g(x)=L .\)

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\), let \(J\) be a closed interval in \(\mathbb{R}\), and let \(c \in J .\) If \(f_{2}\) is the restriction of \(f\) to \(J\), show that if \(f\) has a limit at \(c\) then \(f_{2}\) has a limit at \(c\). Show by example that it does not follow that if \(f_{2}\) has a limit at \(c\), then \(f\) has a limit at \(c\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.