Problem 13 - Precalculus: Mathematics 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 infinity.

Step 1: Identify the Degree of the Polynomial

First, find the degrees of the polynomials in the numerator and the denominator. The numerator is \(x^4\), which has a degree of 4, and the denominator \(1 - x^2 + x^3\), which has a degree of 3.

Step 2: Simplify the Expression by Degree

For large \(x\), the term with the highest degree in both the numerator and the denominator will dominate the others. So, the expression can be simplified by focusing only on the highest degree terms: \(\lim_{x \to \infty} \frac{x^4}{x^3}\).

Step 3: Simplify the Fraction

Simplify the expression by dividing the terms: \(\lim_{x \to \infty} x^{4-3} = \lim_{x \to \infty} x^1 = \lim_{x \to \infty} x\).

Step 4: Determine the Limit

Since \(x\) increases without bound, the limit as \(x\) approaches infinity of \(x\) is infinity.

Key Concepts

Degree of Polynomials

When dealing with polynomials, understanding the degree is crucial. The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial \(x^4\), the degree is 4 because the highest power is 4. Similarly, for a polynomial like \(1 - x^2 + x^3\), you examine each term and identify the highest power, which in this case is 3 (from \(x^3\)). Knowing the degree helps in many calculations, such as determining limits and analyzing the behavior of functions as the variable approaches infinity. It allows us to focus on the most significant term, simplifying our work by honing in on the term that will have the greatest impact as the variable becomes very large or very small.

Simplifying Expressions

Simplifying expressions is a technique used to make complex expressions more manageable. When simplifying, we often focus on the terms with the highest degree, as these dominate the behavior of the polynomial when the variable grows very large.
For example, in the expression \(\frac{x^4}{1-x^2+x^3}\), as \(x\) approaches infinity, the minor terms \(-x^2\) and \(1\) in the denominator become negligible compared to \(x^3\). Hence, it simplifies the expression to \(\frac{x^4}{x^3}\).
This process of focusing on terms with the highest degree allows for easier evaluation of limits while helping to avoid unnecessary complexity.

Infinite Limits

In the world of calculus, infinite limits focus on what happens to a function as the variable approaches infinity. Determining these limits involves understanding the behavior of the terms involved, especially those with the highest degree in polynomials.
When a function like \(\lim_{x \to \infty} \frac{x^4}{x^3}\) is examined, the method used is straightforward because the highest powers in the numerator and denominator will simplify to \(x^{4-3} = x^1\).
As \(x\) tends to infinity, \(x\) becomes enormous; hence the limit is infinity. Infinite limits can give us crucial insights into how functions behave at extreme values and often indicate unbounded growth or decay.