Problem 38 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Simplify the Expression with Dominant Terms

In the given limit problem \( \lim_{x \rightarrow \infty} \frac{x^{2}+2x}{2x^{3}-2x+1} \), observe the degrees of the polynomial in the numerator and the denominator. The highest degree term in the numerator is \(x^2\), and in the denominator, it is \(2x^3\). To simplify, divide each term in both the numerator and the denominator by \(x^3\).

Step 2: Evaluate the Simplified Limit

After dividing each term by \(x^{3}\), the expression becomes \(\lim_{x \rightarrow \infty} \frac{\frac{x^{2}}{x^{3}} + \frac{2x}{x^{3}}}{2-\frac{2x}{x^{3}}+\frac{1}{x^3}} \), which simplifies to \(\lim_{x \rightarrow \infty} \frac{\frac{1}{x} + \frac{2}{x^2}}{2 - \frac{2}{x^2} + \frac{1}{x^3}}\). As \(x\) approaches infinity, the terms with \(\frac{1}{x}, \frac{2}{x^2}\), and \(\frac{1}{x^3}\) tend towards zero.

Step 3: Conclude the Limit Evaluation

Given that \(\frac{1}{x} + \frac{2}{x^2}\) approaches zero and the leading term in the denominator is a constant \(2\), simplify the expression to \(\frac{0}{2}\). Thus, the limit is \(0\).

Key Concepts

Polynomial division

Polynomial division is a mathematical technique that helps us simplify complex expressions by breaking them down into more manageable pieces. It's similar to basic numerical division but applied to algebraic expressions involving variables.

• When we say "polynomial division," we generally refer to dividing one polynomial by another to simplify the expression or to discover another polynomial or remainder.
• For example, dividing each term independently by the highest power involved can help simplify complex rational expressions.

In the exercise, we tackled the limit problem by divides each term in the numerator and the denominator by the highest degree term present in either, which, in this case, was from the denominator: \(x^3\).
This step helps isolate dominant terms and make the expression easier to evaluate as \(x\) approaches infinity.
If you're struggling with polynomial division, practice with basic numerical divisions can be helpful, as the logic is similar.
Remember, understanding the steps behind polynomial division can make seeming complex problems quite manageable with practice.

Dominant terms in polynomials

The concept of dominant terms is pivotal when dealing with polynomials, especially when simplifying expressions or finding limits. The dominant term is the part of the polynomial with the highest degree.

• In any polynomial, the term with the highest exponent of \(x\) is usually the most significant when \(x\) is very large or very small.
• Dominant terms dictate the polynomial's behavior because they grow the fastest as the variable \(x\) increases or decreases indefinitely.

In our example, in the expression \(\frac{x^2 + 2x}{2x^3 - 2x + 1}\), the dominant terms were \(x^2\) in the numerator and \(2x^3\) in the denominator. Recognizing and leveraging dominant terms was key in simplifying the limit expression, allowing us to effectively evaluate it as \(x\) approaches infinity.

Infinite limits

Infinite limits refer to how a function behaves as its input grows larger or approaches a certain point without bound. This concept is crucial for understanding the ultimate behavior of functions and not just their defined values.

• An infinite limit may mean that the function value becomes indefinitely large or small as it approaches the infinite input.
• The expression \(\lim_{x \rightarrow \infty}\) is read as "the limit of a function as \(x\) approaches infinity," which describes this unbounded behavior.

In our scenario, we observed that terms such as \(\frac{1}{x}, \frac{2}{x^2}, \) and \(\frac{1}{x^3}\) tend towards zero as \(x\) increases without bound. These small terms diminish, simplifying the expression to a form where only dominant and constant terms remain influential.
This understanding allows for determining the behavior of the function values at infinity, leading to conclusions such as the function approaching zero or another finite value, offering comprehensive insights into its behavior.