Problem 45 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit. $$\lim _{x \rightarrow \infty} \frac{-4 x^{4}-x^{2}+8}{6 x^{4}-5 x}$$

Step-by-Step Solution

Answer

The limit is \(-\frac{2}{3}\).

Step 1: Identifying Dominant Terms

To solve the limit \( \lim _{x \rightarrow \infty} \frac{-4x^{4} - x^{2} + 8}{6x^{4} - 5x} \), first identify the dominant terms in both the numerator and the denominator. The leading term in the numerator is \(-4x^{4}\), and the leading term in the denominator is \(6x^{4}\). These are the terms with the highest powers of \(x\).

Step 2: Simplifying the Expression

Consider the dominant terms \(\frac{-4x^4}{6x^4}\) which simplifies to \(\frac{-4}{6}\). The other terms become negligible as \(x\) approaches infinity. Hence, the expression simplifies to \(\frac{-2}{3}\) after canceling out the \(x^4\) terms.

Step 3: Calculating the Limit

Now, as \(x\) approaches infinity, the limit of the simplified expression \(\frac{-2}{3}\) remains unchanged. There are no variables left to approach infinity, so the limit is straightforward.

Key Concepts

Understanding Dominant Terms

In calculus, especially when dealing with limits involving polynomials, dominant terms play a crucial role. When analyzing a polynomial for limits, dominant terms are those with the highest power of the variable, usually denoted by "x". These terms are significant because:

• They grow fastest as the variable approaches infinity.
• They largely determine the behavior of the polynomial at extreme values of the variable.

In the given exercise, \( \lim _{x \rightarrow \infty} \frac{-4x^{4} - x^{2} + 8}{6x^{4} - 5x} \), one can see that \(-4x^{4}\) is dominant in the numerator, and \(6x^{4}\) in the denominator. As the variable \(x\) grows indefinitely, the lower power terms like \(-x^{2}\), \(+8\), and \(-5x\) become insignificant compared to the leading \(x^{4}\) terms. Thus, focusing on the dominant terms simplifies the process of determining the limit.

Grasping Infinity in Calculus

Infinity often represents a concept rather than a number in calculus. When working with limits, understanding infinity involves grasping how functions behave as variables increase or decrease without bound.

• In limits, \(x \to \infty\) suggests that \(x\) grows larger and larger.
• Conversely, \(x \to -\infty\) means \(x\) becomes more negative without limit.
• Infinity can help simplify expressions because any constant or lower power term becomes negligible in comparison to infinity.

In the exercise, taking \(\lim _{x \rightarrow \infty} \frac{-4x^{4} - x^{2} + 8}{6x^{4} - 5x}\)involves understanding how these dominant terms behave as \(x\) approaches, but never quite reaches, infinity. By dividing each term by the highest power of \(x\) from the denominator, you essentially zero out terms that don’t have \(x^{4}\), since they become insignificant relative to infinity.

Exploring Rational Expressions

A rational expression in calculus is formed by the division of two polynomials. They are generally expressed as \(\frac{f(x)}{g(x)}\), where both \(f(x)\) and \(g(x)\) are polynomial functions.

• These expressions can sometimes approach certain values at limits.
• Analyzing dominant terms is key to simplifying them, especially as variables tend to infinity.
• They are useful in various calculus concepts such as continuity, asymptotic behavior, and more.

In our example, the rational expression \(\frac{-4x^{4} - x^{2} + 8}{6x^{4} - 5x}\) simplifies primarily because the dominant \(x^{4}\) terms overshadow any lower degree terms. When you divide the numerator and the denominator by \(x^{4}\), you eliminate these dominating powers, leaving a simplified fraction. Hence, understanding the structure of rational expressions often leads to a clearer path to determining the limit.