Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit. $$\lim _{x \rightarrow-\infty} \frac{3 x^{4}-5 x+5}{x-2 x^{2}-x^{4}}$$

Step-by-Step Solution

Answer

The limit is \(-3\).

Step 1: Understanding the Problem

We need to find the limit of the function \( \frac{3x^4 - 5x + 5}{x - 2x^2 - x^4} \) as \( x \to -\infty \). The function is a rational function, so we'll compare the highest degree terms in the numerator and the denominator.

Step 2: Identify Dominant Terms

Observe the terms in the numerator and the denominator. The highest degree term in the numerator is \( 3x^4 \) and in the denominator is \( -x^4 \). As \( x \to -\infty \), we focus on these leading terms because they dominate the behavior of the function.

Step 3: Form the Simplified Expression

The limit simplifies to \( \lim_{x \to -\infty} \frac{3x^4}{-x^4} \). By canceling \( x^4 \), the expression becomes \( \lim_{x \to -\infty} \frac{3}{-1} \).

Step 4: Compute the Limit

The simplified expression \( \frac{3}{-1} \) results in \( -3 \). Therefore, the limit \( \lim_{x \rightarrow -\infty} \frac{3 x^{4}-5 x+5}{x-2 x^{2}-x^{4}} \) evaluates to \(-3\).

Key Concepts

Rational Functions

Rational functions are fractions in which both the numerator and the denominator are polynomials. They are central to calculus because they often arise in mathematical modeling of real-world phenomena. A rational function looks like \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials with real coefficients.

• They can have vertical asymptotes where the denominator equals zero.
• Horizontal or oblique asymptotes may exist depending on the degrees of the numerator and the denominator.

It's important to understand the behavior of these functions as the input \( x \) values become very large—positively or negatively. Limits are a useful tool in this analysis because they help determine the end behavior of these functions. For the given problem, we can analyze the limit \( \lim_{x\rightarrow -\infty} \frac{3x^4 - 5x + 5}{x - 2x^2 - x^4} \) to understand how the function behaves as \( x \) approaches negative infinity.

Dominant Terms

When examining limits of rational functions, identifying the dominant or leading terms is crucial. These are the terms with the highest power of \( x \) in both the numerator and the denominator. As \( x \to \pm \infty \), these terms grow much faster than the others, significantly influencing the behavior of the entire function.In the example function \( \frac{3x^4 - 5x + 5}{x - 2x^2 - x^4} \):

• The highest degree term in the numerator is \( 3x^4 \).
• The highest degree term in the denominator is \( -x^4 \).

For large values of \( x \), the lower degree terms like \(-5x\), \(5\), or even \(-2x^2\) become negligible. Focusing on the dominant terms simplifies the limit calculations as other terms tend to zero compared to the highest degree terms. This approach is key in simplifying the function to reach a solution.

Simplifying Expressions

Simplifying expressions, especially when dealing with limits, allows us to focus on how the most influential terms behave. This is done by dividing each term of the rational function by the highest power of \( x \) present in the denominator or numerator. This highlights the dominant terms and disregards less significant ones.For the example problem, we can simplify \( \frac{3x^4 - 5x + 5}{x - 2x^2 - x^4} \) by dividing each term by \( x^4 \), which is the dominant term's degree:

• Numerator becomes \( \frac{3x^4}{x^4} - \frac{5x}{x^4} + \frac{5}{x^4} = 3 - \frac{5}{x^3} + \frac{5}{x^4} \).
• Denominator becomes \( \frac{x}{x^4} - \frac{2x^2}{x^4} - \frac{x^4}{x^4} = \frac{1}{x^3} - \frac{2}{x^2} - 1 \).

As \( x \to -\infty \), terms containing \( \frac{1}{x^n} \) where \( n > 0 \) approach zero. Thus, the expression simplifies further to \( \frac{3}{-1} = -3 \). Simplifying the expression not only helps solve the limit but also clarifies how the function behaves at extremes.