Problem 8 - Precalculus: Mathematical for Calculus

Question

Find the limit. $$\lim _{x \rightarrow-\infty} \frac{x^{2}+2}{x^{3}+x+1}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Analyze the Degree of Polynomials

Identify the degree of the polynomial in the numerator and the denominator. The degree of the numerator \( x^2 + 2 \) is 2, and the degree of the denominator \( x^3 + x + 1 \) is 3.

Step 2: Divide by the Highest Degree Term in the Denominator

To simplify the expression, divide every term in both the numerator and denominator by \( x^3 \), the highest degree term in the denominator. This gives us: \(\lim_{x \to -\infty} \frac{\frac{x^2}{x^3} + \frac{2}{x^3}}{\frac{x^3}{x^3} + \frac{x}{x^3} + \frac{1}{x^3}} = \lim_{x \to -\infty} \frac{\frac{1}{x} + \frac{2}{x^3}}{1 + \frac{1}{x^2} + \frac{1}{x^3}}.\)

Step 3: Evaluate the Limits of Each Term as \( x \to -\infty \)

Consider each term separately as \( x \to -\infty \). \(\frac{1}{x} \to 0, \frac{2}{x^3} \to 0, \frac{1}{x^2} \to 0, \frac{1}{x^3} \to 0.\) The expression simplifies to: \(\lim_{x \to -\infty} \frac{0 + 0}{1 + 0 + 0} = 0.\)

Step 4: Conclude the Limit

Based on the simplification, the limit of the function as \( x \to -\infty \) is 0.

Key Concepts

Polynomial Degree Analysis

In the realm of calculus, particularly when finding the limit of a function, understanding the degree of polynomials takes center stage.
By identifying the degree, we gain insights into which terms dominate the behavior of the polynomial as variables approach extremes like infinity.
Here's how you can approach polynomial degree analysis:

• The degree of a polynomial is determined by the term with the highest exponent.
• For the polynomial in the numerator, \(x^2 + 2\), the highest exponent is 2, indicating a degree of 2.
• For the polynomial in the denominator, \(x^3 + x + 1\), the degree is governed by \(x^3\), hence the degree is 3.

Recognizing these degrees is crucial because they tell us which terms to focus on when simplifying the function.
In limits involving infinity, lower-degree terms become negligible compared to higher-degree terms.

Asymptotic Behavior

Asymptotic behavior describes how functions act as they approach a specific boundary or point, such as infinity.
This is especially important when analyzing how the terms in a function affect the overall expression.

• When encountering a limit problem that involves infinity, we observe how the terms behave as \(x\) approaches an extreme value.
• The main goal is to reduce complexity by removing terms that diminish to zero, especially when comparing different degrees.
• In our exercise, analyze \(\frac{x^2+2}{x^3+x+1}\) as \(x \to -\infty\). The lower-degree terms in both the numerator and denominator trend towards zero.

This step of eliminating insignificant terms helps simplify the problem.
It refines our focus on the dominant behavior of the polynomial near infinity, leading us to more easily evaluate its limit.

Infinity Limits

Infinity limits help mathematicians understand a function's behavior as a variable grows endlessly large, in either the positive or negative direction.
These calculations reveal how functions stabilize, diverge, or approach specific values.

• The step-by-step solution divides each term by \(x^3\), the highest power in the denominator, simplifying the form to \(\frac{\frac{1}{x} + \frac{2}{x^3}}{1 + \frac{1}{x^2} + \frac{1}{x^3}}\).
• As \(x\) grows more negative, terms like \(\frac{1}{x}\) and \(\frac{2}{x^3}\) tend toward zero, due to their negative powers in the fraction.
• The constant term 1 in the denominator remains, while the rest vanish due to the overwhelming influence of negative infinity.

The ultimate result here, as we simplify, is a conclusion that the original expression evaluates to 0 as \(x\) approaches \(-\infty\).
Infinity limits simplify our understanding of how expressions resolve in extreme cases.