Problem 6 - Precalculus Mathematics for Calculus

Question

Find the limit. $$\lim _{x \rightarrow \infty} \frac{3}{x^{4}}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Problem

The exercise requires us to find the limit of the function \( \frac{3}{x^4} \) as \( x \) approaches infinity. This means we are looking to evaluate the behavior of the expression \( \frac{3}{x^4} \) when \( x \) becomes very large.

Step 2: Analyze the Denominator

As \( x \) approaches infinity, the denominator \( x^4 \) becomes very large. Since \( x^4 \) is growing to infinity, it heavily influences the value of the entire fraction \( \frac{3}{x^4} \).

Step 3: Evaluate the Expression

For very large \( x \), the expression \( \frac{3}{x^4} \) can be viewed as \( \frac{3}{\text{something very large}} \). As \( x^4 \) increases, \( \frac{3}{x^4} \) becomes very small, approaching zero.

Step 4: Conclude the Limit

Since \( \frac{3}{x^4} \rightarrow 0 \) as \( x \rightarrow \infty \), the limit of \( \frac{3}{x^4} \) as \( x \) approaches infinity is 0.

Key Concepts

Infinity

When we discuss "infinity" in terms of limits and functions, we're exploring what happens to a function's value as the input grows without bound. To think about infinity, imagine something that keeps going and going, never stopping or reaching a maximum point. Similarly, approaching infinity usually means continually increasing the input value. In mathematics, we often write this concept as \( x \to \infty \) for functions involving \( x \).

• When a function's input tends to infinity, it can lead to various outcomes, like getting closer to a specific number or growing without any limitation.
• Understanding this concept helps in grasping how functions behave at their extremes, beyond the usual range.

Understanding limits at infinity is a crucial part of calculus. It helps us predict the behavior of functions far beyond specific numerical computations.

Asymptotic Behavior

Asymptotic behavior refers to how a function acts as it nears a particular line or shape, usually at the extremes (like infinity). This behavior tells us how the function "hugs" or approaches an axis or line without touching it.For the function \( \frac{3}{x^4} \), as \( x \to \infty \), its asymptotic behavior shows that it approaches zero, although never quite reaching it fully. Here's why:

• As \( x \) increases, the denominator \( x^4 \) becomes colossal, making \( \frac{3}{x^4} \) shrink dramatically.
• The number 3 stays constant while \( x^4 \) grows rapidly, emphasizing the diminishing effect.

When examining graphs, the x-axis acts as a horizontal asymptote, meaning our function gets exceedingly close to this axis but never intersects it as \( x \) heads to infinity. Asymptotic behavior is significant in understanding limits and predicting long-term behavior.

Rational Functions

Rational functions are the quotient of two polynomial functions. The function \( \frac{3}{x^4} \) is a simple example of a rational function, where the numerator is a constant and the denominator is a polynomial of degree 4.

• Rational functions often have characteristic features such as asymptotes and bounded behavior for extreme values of \( x \).
• They commonly appear in calculus problems involving limits because they illustrate diverse outcomes as \( x \) moves towards infinity or negative infinity.

In our exercise, the rational function \( \frac{3}{x^4} \) highlights a significant concept: when a higher-degree polynomial is in the denominator, the function's value tends toward zero as \( x \) becomes extremely large.Learning about rational functions allows students to understand a wide array of mathematical scenarios, from basic limit problems to intricate real-world models involving rates and proportions.