Problem 11 - Precalculus: Mathematical for Calculus

Question

Find the limit. $$\lim _{x \rightarrow \infty} \frac{x^{4}}{1-x^{2}+x^{3}}$$

Step-by-Step Solution

Answer

The limit is \(\infty\).

Step 1: Identify the dominant term in the numerator and denominator

First, identify the terms in both the numerator and the denominator with the highest powers of \(x\). The dominant term in the numerator is \(x^4\), and in the denominator, the dominant term is \(x^3\).

Step 2: Rewrite the expression by factoring out dominant terms

Factor \(x^3\) out of the denominator to simplify the expression: \(\lim _{x \rightarrow \infty} \frac{x^{4}}{x^3(\frac{1}{x^3} - \frac{x^2}{x^3} + 1)}\).

Step 3: Simplify the expression

Cancel out the \(x^3\) term from the numerator and denominator, resulting in \(\lim _{x \rightarrow \infty} x \cdot \frac{1}{\frac{1}{x^3} - \frac{1}{x} + 1}\).

Step 4: Evaluate the limit as x approaches infinity

As \(x\) approaches infinity, the terms \(\frac{1}{x^3}\) and \(\frac{1}{x}\) approach zero. Therefore, the expression simplifies to \(\lim _{x \rightarrow \infty} x \cdot \frac{1}{1} = x\). The limit of \(x\) as \(x\) approaches infinity is \(\infty\).

Key Concepts

Dominant Term

Understanding the concept of a dominant term is essential when dealing with limits, especially when approaches involve large values such as infinity. The dominant term in a polynomial is the term that grows the fastest as the variable approaches infinity. Let's consider an expression like \(x^4 + 2x^3 - x + 7\). In this case, the dominant term is \(x^4\) since it is raised to the highest power. When calculating limits, it is particularly helpful to identify the dominant term because it generally dictates the behavior of the entire function as \(x\) approaches infinity or negative infinity.

Dominant terms affect both the numerator and the denominator of ratios. For example:

• In \(\lim_{x \rightarrow \infty} \frac{x^4}{1-x^2+x^3}\), the dominant term in the numerator is \(x^4\), and in the denominator, it is \(x^3\).
• Recognizing these terms helps simplify complex expressions and focus on the parts of the function that truly matter in the limit calculations.

By examining the dominant terms, we can often predict the behavior of a function graphically and numerically, aiding in the understanding of a function's asymptotic behavior.

Factorization

Factorization plays a critical role in simplifying expressions, particularly when finding the limits of functions as the variable approaches infinity. By breaking down complex expressions through factorization, we can make calculations more manageable and less error-prone. This process can often reveal the dominant terms we discussed earlier.

For instance, if we have the expression \(1-x^2+x^3\), we can factor out the highest power of \(x\) in the denominator—\(x^3\), resulting in:

• \( \frac{x^4}{x^3(\frac{1}{x^3} - \frac{x^2}{x^3} + 1)} \)
• This transforms our expression into a simpler form, where the original complex terms are expressed as fractions or constants.

This step of factorization allows us to effectively cancel out terms when they appear in both the numerator and denominator. In our example:

• The \(x^3\) term is factored from the denominator and canceled with part of the numerator's \(x^4\), leaving \(x\).

Factorization not only simplifies algebraic expressions but also reveals key insights into the expression's behavior as it tends to infinity.

Infinity

Infinity is a crucial concept in calculus and mathematics as a whole. It describes something without any bound or end. When we calculate limits, especially towards infinity, we're interested in what happens to a function as the variable grows larger and larger. This helps us understand the behavior of functions comprehensively.

In the given problem, \( \lim_{x \rightarrow \infty} \frac{x^4}{1-x^2+x^3} \), we evaluate how the function behaves as \(x\) approaches infinity. The step-by-step solution simplifies the expression to \( \lim_{x \rightarrow \infty} x \), which reveals that the function increases without bound as \(x\) becomes very large.

Key points regarding infinity in limits include:

• Main terms dominate: Overwhelmingly, only the most significant terms affect the outcome. Smaller terms vanish in importance.
• Understanding asymptotic behavior: This concept helps in analyzing the end-behavior of functions, providing insight into horizontal and vertical asymptotes.

Conclusively, approaching limits at infinity gives us vital insight into the infinite nature of mathematical functions, helping us plot the course of these mathematical journeys.