Problem 31 - Calculus: Early Transcendentals

Question

Find the limit or show that it does not exist. \( \displaystyle \lim_{x \to \infty} \frac{x^4 - 3x^2 + x}{x^3 - x + 2} \)

Step-by-Step Solution

Answer

The limit is \(+\infty\).

Step 1: Identify the degrees

Observe the degrees of the polynomials in the numerator and the denominator. The numerator, \( x^4 - 3x^2 + x \), is of degree 4. The denominator, \( x^3 - x + 2 \), is of degree 3.

Step 2: Determine the highest power comparison

To evaluate the limit as \( x \to \infty \), compare the highest powers of \( x \) in the numerator and denominator. Since the highest power in the numerator (4) is greater than the highest power in the denominator (3), the limit of the expression will tend towards infinity.

Step 3: Simplification check

Since the degree in the numerator is greater than in the denominator, simplification or factoring won't change the behavior towards infinity. Simplifying by the highest degree in the denominator won't nullify the dominant term \( x^4 \) in the numerator.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables raised to whole number powers, combined using addition, subtraction, and multiplication. They are typically written in standard form, where terms are ordered from the highest degree to the lowest.
For example, in the expression \(x^4 - 3x^2 + x\), the terms are arranged based on the descending powers of \(x\). Each term in a polynomial has a coefficient, which is the number in front of the variable.
Understanding polynomial functions is important because they show up frequently in calculus, especially when investigating limits or analyzing the behavior at infinity. They also serve as an introduction to more complex functions, making grasping the fundamental concepts crucial for further study.

Degree of Polynomials

The degree of a polynomial is defined as the highest power of the variable present in the polynomial. It plays a crucial role in determining the behavior of the polynomial function, particularly when assessing limits and end behavior.
In the polynomial \(x^4 - 3x^2 + x\), the degree is 4 because \(x^4\) is the term with the highest exponent. Similarly, for \(x^3 - x + 2\), the degree is 3 due to the \(x^3\) term.
Knowing the degree helps in predicting the behavior of a polynomial as \(x\) approaches infinity. When comparing two polynomial functions, the function with the higher degree dominates the other. This is essential when analyzing limits, as seen in the original exercise.

Behavior at Infinity

The behavior of polynomial functions as \(x\) approaches infinity is essential in understanding limits. This behavior is primarily determined by the highest degree term in the polynomial. Regardless of the other terms, the term with the highest power of \(x\) will dominate.
For example, when evaluating \(\lim_{x \to \infty} \frac{x^4 - 3x^2 + x}{x^3 - x + 2}\), it's crucial to compare the degrees of the numerator and denominator. Here, the numerator has a higher degree (4) compared to the denominator (3). As a result, the ratio of these two will increase without bounds, and the limit tends towards infinity.
This concept helps in predicting the end behavior of the function without actually calculating the limit, simply by assessing the dominant high-degree terms.