Problem 12 - Calculus with Concepts in Calculus

Question

Find the given limit. $$ \lim _{x \rightarrow-\infty} \frac{2-3 x-4 x^{2}}{3 x^{2}+6 x+10} $$

Step-by-Step Solution

Answer

The limit is \(-\frac{4}{3}\).

Step 1: Identify the Dominant Terms

For limits as x approaches infinity or negative infinity, the behavior of the rational function is determined by the highest degree terms in the numerator and the denominator. Here, the dominant terms are \(-4x^2\) in the numerator and \(3x^2\) in the denominator.

Step 2: Simplify the Fraction

Focus on the dominant terms: \( \frac{-4x^2}{3x^2} \). We can simplify this expression to \( \frac{-4}{3} \). All other terms become negligible compared to these as \(x\) approaches negative infinity.

Step 3: Apply the Limit

As \(x\) approaches \(-\infty\), the other terms in the original expression \((2, -3x, 6x, 10)\) become insignificant, and the limit of \( \frac{2 - 3x - 4x^2}{3x^2 + 6x + 10} \) essentially reduces to \( \frac{-4}{3} \).

Key Concepts

Rational Functions

Rational functions are expressions built from a division of two polynomials. They have the general form \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials. Understanding these functions is crucial in calculus, particularly when working with limits. By examining the highest-degree terms within the numerator and denominator, we can determine the behavior of the function as \(x\) approaches specific values like infinity or negative infinity. This method helps us simplify complex functions and analyze their trends. For example, in the given exercise, the rational function is \(\frac{2 - 3x - 4x^2}{3x^2 + 6x + 10} \). Here, identifying the leading terms in both the numerator and denominator is a powerful strategy to simplify the function and understand its asymptotic behavior.

Dominant Terms

Dominant terms refer to the terms in a polynomial with the highest degree. In the context of limits, especially with rational functions, focusing on these terms can greatly simplify our calculations. By doing so, the influence of lower-degree terms becomes negligible as \(x\) tends to infinity or negative infinity. This means:

• The highest-degree terms dictate the behavior of the function.
• These terms allow us to simplify the overall expression effectively.

In the original problem, the dominant terms were \(-4x^2\) in the numerator and \(3x^2\) in the denominator. By isolating these, we obtain a simpler rational function \(\frac{-4x^2}{3x^2}\), which reduces to a constant value. This method provides a clearer understanding of the limit of the original function as \(x\) approaches negative infinity.

Infinity Limits

Infinity limits in calculus deal with the behavior of functions as the input approaches extremely large values or negative values, like \(\infty\) or \(-\infty\). This concept is crucial for finding out how functions behave asymptotically. In our exercise, we are tasked with finding the limit as \(x\) approaches \(-\infty\). Some steps to consider include:

• Identifying which terms in the function have the greatest influence at extreme values of \(x\).
• Simplifying the function using these dominant terms.
• Calculating the limit based on the simplified expression.

For the provided rational function, after simplifying to the expression \(\frac{-4}{3}\), we know as \(x\) approaches \(-\infty\), the limit converges to \(\frac{-4}{3}\). This approach highlights how infinity limits allow us to understand the far-reaching behavior of functions effectively, leading to more insightful analysis.