Problem 10 - Precalculus: Mathematical for Calculus

Question

Find the limit. $$\lim _{r \rightarrow \infty} \frac{4 r^{3}-r^{2}}{(r+1)^{3}}$$

Step-by-Step Solution

Answer

The limit is 4.

Step 1: Identify the Highest Power of r in the Numerator and Denominator

In the expression \( \frac{4 r^{3}-r^{2}}{(r+1)^{3}} \), identify the highest power of \( r \) in the numerator and the denominator. In the numerator \( 4r^3 - r^2 \), the highest power is \( r^3 \). In the denominator \( (r+1)^3 \), we expand it to get \( r^3 + 3r^2 + 3r + 1 \), and the highest power is also \( r^3 \).

Step 2: Divide Each Term by the Highest Power

To simplify the limit, divide each term in the fraction by \( r^3 \), which is the highest power identified. This gives: \( \frac{4 - \frac{1}{r}}{1 + \frac{3}{r} + \frac{3}{r^2} + \frac{1}{r^3}} \).

Step 3: Evaluate the Limit as r Approaches Infinity

As \( r \to \infty \), all terms containing \( \frac{1}{r} \), \( \frac{1}{r^2} \), and \( \frac{1}{r^3} \) approach zero. This simplifies the expression to \( \frac{4}{1} \).

Step 4: Calculate the Final Limit

Simplify the expression: \( \frac{4}{1} = 4 \). Thus, the limit is \( 4 \).

Key Concepts

Rational Functions

Rational functions are a special type of function defined by a fraction of polynomials. They have the general form \( f(x) = \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials. Unlike some other functions, rational functions might not be defined everywhere. They are undefined where the denominator \(Q(x)\) equals zero. These points often lead to vertical asymptotes, which are lines the graph of the function approaches but never touches.

In this exercise, the rational function is \( \frac{4r^3 - r^2}{(r+1)^3} \). The numerator and the denominator both involve polynomials. Notice that as \(r\) becomes very large, or approaches infinity, rational functions can behave more predictably. By understanding the behavior of the highest powers in the numerator and denominator, like \(r^3\), we can determine how the function behaves at extreme values of \(r\).

• Rational functions can often be simplified by factoring and canceling.
• The limit at infinity often depends on the highest degree terms in numerator and denominator.

Infinite Limits

Infinite limits describe the behavior of a function as a variable approaches infinity or negative infinity. Rather than finding a specific value, infinite limits help us understand the trend or approach of a function's graph.

In this exercise, as \( r \rightarrow \infty \), we are primarily interested in how the terms behave that dominate the function. To understand infinite limits in rational functions, such as \( \lim_{r \to \infty} \frac{4r^3 - r^2}{(r+1)^3} \), we focus on the highest degree terms. By dividing each term by \(r^3\), the highest power of \(r\), smaller, negligible terms approach zero. This simplifies our calculations substantially. Understanding which terms "go to zero" allows us to see that even though there are many terms involved, ultimately the chosen limit simplifies significantly.

• Infinite limits help evaluate the behavior as variables grow very large or very small.
• Simplification is a key step to finding most infinite limits.

Polynomial Division

Polynomial division is a useful algebraic tool for breaking down complex polynomials and simplifying rational expressions. When dealing with limits involving rational functions, polynomial division can guide us in making simplifications.

In this step-by-step solution, polynomial division is implicitly used when identifying and dividing each term by the highest power of \(r\) present. By doing so, we instantly prioritize the most significant terms relative to \(r\)'s growth or shrinkage. Here, dividing by \(r^3\) isolates the leading coefficients and allows other terms to diminish towards zero as \(r\) approaches infinity. This calculation shows how polynomial division can simplify seemingly complex fractions to determine behavior at extremes.

• Dividing by the highest power in polynomials can significantly simplify functions.
• Understanding polynomial division aids in effective calculation especially in limits.