Problem 47 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Dominant Terms

Examine the numerator and denominator of the function \( \frac{2x^2 - 1}{3x^4 + 2} \). The dominant term in the numerator is \(2x^2\) and in the denominator is \(3x^4\). These terms will primarily dictate the behavior as \(x\) approaches infinity.

Step 2: Rewrite the Expression

Factor out the dominant terms in both the numerator and denominator. Rewrite the function as: \[\frac{2x^2(1 - \frac{1}{2x^2})}{3x^4(1 + \frac{2}{3x^4})}\] This simplification will help make the limit clearer.

Step 3: Simplify the Fraction

Further simplify the expression: \[\frac{2x^2}{3x^4} \cdot \frac{1 - \frac{1}{2x^2}}{1 + \frac{2}{3x^4}}\] This reduces to:\[\frac{2}{3x^2} \cdot \frac{1 - \frac{1}{2x^2}}{1 + \frac{2}{3x^4}}\]

Step 4: Evaluate the Limit as \(x \to \infty\)

Consider each part of the fraction as \(x\) approaches infinity. The \(\frac{2}{3x^2}\) term approaches zero because the denominator grows much faster than the numerator. Additionally, the terms \(\frac{1}{2x^2}\) and \(\frac{2}{3x^4}\) also approach zero as \(x\to \infty\).

Step 5: Conclude the Limit

As \(x\to \infty\), the expression simplifies to \(0 \cdot \frac{1 - 0}{1 + 0} = 0\). Therefore, the limit is \(0\).

Key Concepts

Dominant Terms

In calculus, when evaluating limits of functions as the variable approaches infinity, identifying the dominant terms is essential. The dominant terms in both the numerator and denominator are the terms with the highest power of the variable. These terms largely influence the behavior of the entire expression for very large values of the variable.

• The expression given here is \( \frac{2x^2 - 1}{3x^4 + 2} \).
• The **dominant term** in the numerator is \(2x^2\).
• The **dominant term** in the denominator is \(3x^4\).

By focusing on these dominant terms, you can effectively predict the function's behavior as \(x\) approaches infinity.

Infinite Limits

Infinite limits involve determining the behavior of a function as the variable approaches a very large value, often written as \(x \rightarrow \infty\). In this context, a limit evaluates to infinity if the numerator grows faster than the denominator, or it might evaluate to zero if the denominator grows faster.In the expression \(\frac{2x^2 - 1}{3x^4 + 2}\), the denominator has a higher power (4) compared to the numerator (2). This means the denominator grows much faster than the numerator as \(x\) becomes very large. Thus, the overall fraction tends towards zero.

Understanding infinite limits is crucial for predicting how functions behave as they stretch towards larger values, which often leads to simplifying the original problem into something more manageable.

Numerator and Denominator Analysis

A detailed analysis of both the numerator and the denominator helps to simplify complex rational expressions. This involves breaking down each part to see which components have the strongest impact.For the expression \(\frac{2x^2 - 1}{3x^4 + 2}\):

• The numerator: \(2x^2 - 1\) is dominated by the \(2x^2\) term.
• The denominator: \(3x^4 + 2\) is dominated by the \(3x^4\) term.

By factoring these dominant terms, we can rewrite the expression. This assists in clarifying how each part behaves as \(x\) approaches infinity, further easing the process of limit evaluation.

Simplifying Fractions

Simplifying fractions involves reducing expressions so they are easier to analyze, especially in limit evaluation. This process includes extracting and dividing out common factors, and rewriting each component in its simplest form.In our limit problem, we start by factoring out the dominant terms:\[\frac{2x^2 (1 - \frac{1}{2x^2})}{3x^4 (1 + \frac{2}{3x^4})}\]This reduces to:\[\frac{2}{3x^2} \cdot \frac{1 - \frac{1}{2x^2}}{1 + \frac{2}{3x^4}}\]As \(x\) approaches infinity:

• \(\frac{2}{3x^2}\) tends to zero because the power in the denominator is larger.
• The term \(1 - \frac{1}{2x^2}\) simplifies to 1 because \(\frac{1}{2x^2}\) approaches zero.
• The term \(1 + \frac{2}{3x^4}\) simplifies to 1 likewise.

Therefore, the simplified result of the limit as \(x \to \infty\) is \(0\). Simplifying such expressions makes evaluating the original limit far more straightforward and intuitive.