Problem 25 - Single Variable Calculus

Question

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

Step-by-Step Solution

Answer

The limit of the given expression as \(x\) approaches infinity is \(\frac{1}{9}\).

Step 1: Identify the common denominator

The given expression involves two fractions: \(\frac{x^3}{3x^2 - 2}\) and \(\frac{x^2}{3x + 1}\). To combine these fractions, we need to identify the common denominator, which for these two fractions is \((3x^2 - 2)(3x + 1)\).

Step 2: Combine the fractions

To combine the two fractions, rewrite each fraction in terms of the common denominator. This will allow us to combine them into one fraction: \[ \frac{x^3}{3x^2 - 2} - \frac{x^2}{3x + 1} = \frac{x^3(3x + 1)}{(3x^2 - 2)(3x + 1)} - \frac{x^2(3x^2 - 2)}{(3x^2 - 2)(3x + 1)} \] Now combine the two fractions into one: \[ \frac{x^3(3x + 1) - x^2(3x^2 - 2)}{(3x^2 - 2)(3x + 1)} \]

Step 3: Simplify the numerator

Expand and simplify the numerator of the combined fraction: \[ x^3 (3x + 1) - x^2 (3x^2 - 2) = 3x^4 + x^3 - 3x^4 + 2x^2 = x^3 + 2x^2 \] So now the expression is: \[ \frac{x^3 + 2x^2}{(3x^2 - 2)(3x + 1)} \]

Step 4: Find the limit as x approaches infinity

To find the limit of this expression as x approaches infinity, divide both the numerator and the denominator by the highest power of x in the denominator, which in this case is \(x^2\). This results in: \[ \frac{\frac{x^3}{x^2} + \frac{2x^2}{x^2}}{\frac{(3x^2 - 2)(3x + 1)}{x^2}} = \frac{x + 2}{(3 - \frac{2}{x^2})(3x + \frac{1}{x})} \] As x approaches infinity, the terms \(\frac{2}{x^2}\) and \(\frac{1}{x}\) approach 0. Therefore, the expression approaches: \[ \frac{x + 2}{(3 \cdot 3x)} = \frac{x + 2}{9x} \]

Step 5: Find the limit as x approaches infinity of the simplified expression

As x approaches infinity, the expression \(\frac{x + 2}{9x}\) approaches: \[ \lim_{x \rightarrow \infty} \frac{x + 2}{9x} = \lim_{x \rightarrow \infty} \frac{1 + \frac{2}{x}}{9} = \frac{1}{9} \] So the limit of the given expression as x approaches infinity is \(\frac{1}{9}\).

Key Concepts

Infinity Limits

In calculus, when we talk about limits as a variable approaches infinity, we are investigating how a function behaves as its input goes towards very large values. This concept is crucial, especially when dealing with functions that grow without bound. Understanding infinity limits helps us predict the behavior of functions at extremely large values, which is a fundamental part of analyzing mathematical models.

When evaluating limits as \(x\) approaches infinity, we often aim to see which part of the function dominates as the input grows. For a polynomial, the term with the highest degree generally governs the behavior since it increases much faster compared to other terms. This means that if you have a polynomial in the numerator and denominator, each with different highest power terms, focusing on these leading terms allows simplification of the limit evaluation.

• Helps assess function behavior at large inputs.
• Determines how functions grow as inputs grow.
• Focuses on dominant terms for simplification.

This insight is highly beneficial for computing limits at infinity efficiently, as demonstrated in our problem, where the limit simplifies to \(\frac{1}{9}\).

Rational Functions

Rational functions are expressions resulting from the division of two polynomials. They show up frequently in both pure mathematics and applied contexts like physics and engineering. The function in our exercise is indeed a rational function, as it is a division of polynomials.

• Formed by dividing two polynomials.
• Common in mathematical modeling.
• Feature asymptotic behavior at infinities.

Rational functions often exhibit asymptotic behavior because the degrees of polynomials in the numerator and denominator dictate how these functions behave as \(x\) approaches very large or small values. In our exercise, combining fractions involved operations with these polynomials, followed by simplification.

The degree of the numerator and denominator plays a crucial role in determining the limit at infinity. If the degree of the numerator is less than that of the denominator, like our example, the rational function will tend towards zero as \(x\) heads to infinity. However, when degrees are equal or the numerator’s degree is higher, the behavior may be different, requiring different analysis techniques.

Limit Simplification

Simplifying expressions is a significant step in computing limits effectively and efficiently. In our exercise, the expression involved was complex, with two rational functions. To deal with such complexity, combining fractions over a common denominator was a necessary process. This step allowed simplification before taking the limit.

• Combines fractions to simplify expressions.
• Uses leading terms for easier calculations.
• Enables quick evaluation of limits.

After simplifying, dividing both the numerator and the denominator by the highest power of \(x\) present in the denominator simplifies the process further. It eases understanding by focusing on dominant terms impacting the expression's growth as \(x\) becomes large.
For our example, simplification led us to analyze \( \frac{x + 2}{9x} \), transforming it into a much simpler expression \( \frac{1 + \frac{2}{x}}{9} \), which made calculating the limit straightforward. This approach allows us to quickly determine the behavior of polynomials' division at infinity, emphasizing the power of simplification in calculus.