Problem 46 Determine whether each limit is ... [FREE SOLUTION]

Chapter 3: Problem 46

Determine whether each limit is equal to $\infty$ or $-\infty$. $$\lim _{x \rightarrow \infty} x^{3}+x$$

Short Answer

Expert verified

The limit is $ \infty $.

Step by step solution

Identify the dominant term

As we examine the polynomial expression $x^3 + x$, we identify that the term $x^3$ is of a higher degree than $x$. Therefore, as $x \rightarrow \infty$, $x^3$ will grow much faster than $x$.

Analyze the dominant term

Since $x^3$ is the dominant term and it is positive, observe the behavior of $x^3$ as $x \rightarrow \infty$. It increases to infinity.

Determine the limit

By recognizing that the dominant term $x^3$ dictates the behavior of the entire polynomial as $x \rightarrow \infty$, we conclude that the limit of $x^3 + x$ is also infinity.

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.

dominant term

When evaluating limits at infinity, especially for polynomial expressions, identifying the **dominant term** is crucial. The dominant term is the term in the polynomial that grows the fastest as the variable approaches infinity. In other words, it's the term with the highest power. For example, in the polynomial expression $x^3 + x$, the dominant term is $x^3$ because it has the highest exponent. As we analyze the behavior of this dominant term as $x$ increases, we can make predictions about the entire polynomial.

polynomial behavior

Understanding **polynomial behavior** is essential in limit evaluation at infinity. A polynomial's behavior as its variable approaches infinity is heavily influenced by its dominant term. In our example, $x^3 + x$, the polynomial will behave similarly to its dominant term $x^3$ because $x^3$ grows much faster than the linear term $x$. Therefore, you can almost 'ignore' the lower-degree terms in the polynomial when determining the limit at infinity. This simplification allows you to focus on how the dominant term controls the overall growth of the polynomial.

limit evaluation

Once you have identified the dominant term and understood the polynomial's behavior, you can proceed to **limit evaluation**. To determine the limit of $x^3 + x$ as $x$ approaches infinity, we rely on the understanding that the dominant term $x^3$ drives the polynomial's growth. As $x$ tends toward infinity, $x^3$ increases without bound. Consequently, the entire polynomial expression $x^3 + x$ also increases without bound. Therefore, the limit of $x^3 + x$ as $x$ approaches infinity is $\infty$. This straightforward process of identifying the dominant term and examining its behavior helps simplify the complex evaluation of polynomial limits at infinity.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Determine whether each limit is equal to \(\infty\) or \(-\infty\). $$\lim _{x \rightarrow \infty} x^{3}+x$$

Short Answer

Step by step solution

Identify the dominant term

Analyze the dominant term

Determine the limit

Key Concepts

dominant term

polynomial behavior

limit evaluation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Geometry

Probability and Statistics

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.