Problem 12 - Precalculus: Mathematics 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: Understanding the Problem

We are tasked to find the limit as \( r \) approaches infinity of the expression \( \frac{4r^{3} - r^{2}}{(r+1)^3} \). The strategy is typically to simplify the expression to see its behavior at large \( r \).

Step 2: Simplifying the Expression

First, expand the denominator. \( (r+1)^{3} = r^{3} + 3r^{2} + 3r + 1 \). Our expression becomes: \( \frac{4r^{3} - r^{2}}{r^{3} + 3r^{2} + 3r + 1} \).

Step 3: Finding the Dominant Terms

Identify the dominant terms in the numerator and the denominator. In the numerator, \( 4r^3 \) dominates \( -r^2 \), and in the denominator, \( r^3 \) is the dominant term.

Step 4: Simplifying the Limit Expression

Since \( r^3 \) is the dominant term in both the numerator and the denominator, divide each term by \( r^3 \) to simplify the expression: \( \lim _{r \rightarrow \infty} \frac{4 - \frac{1}{r}}{1 + \frac{3}{r} + \frac{3}{r^2} + \frac{1}{r^3}} \).

Step 5: Evaluating the Limit

As \( r \) approaches infinity, terms involving \( \frac{1}{r} \), \( \frac{1}{r^2} \), and \( \frac{1}{r^3} \) approach zero. Thus, the expression simplifies to \( \frac{4 - 0}{1 + 0 + 0 + 0} = 4 \).

Key Concepts

Dominant Terms

In calculus, when working with limits, identifying dominant terms is a crucial step. These are the terms that have the greatest impact on the expression as the variable, in this case, \( r \), approaches infinity.

• Dominant terms help us focus on the parts of the expression that truly affect the limit's behavior.
• For polynomial expressions like \( 4r^3 - r^2 \), the term with the highest degree, here \( 4r^3 \), will dominate.
• Similarly, in the denominator \((r+1)^3\), expanding gives \( r^3 + 3r^2 + 3r + 1 \), where \( r^3 \) is the dominant term.

By concentrating on these terms, you can often simplify the calculation of limits for functions as the variable approaches infinity.

Simplifying Expressions

Simplifying expressions before evaluating a limit can make the process significantly easier. You'll want to break down complex expressions into simpler forms.

• Begin by expanding any parts of the expression that are not in a simple polynomial form, such as \((r+1)^3\), which expands to \( r^3 + 3r^2 + 3r + 1 \).
• Once expanded, compare the degree of terms across the numerator and the denominator to determine what can be simplified.
• Identify common factors and divide them out. In our case, dividing all terms by \( r^3 \) works well since it’s the dominant term.

This results in a much friendlier form for limit evaluation: \( \frac{4 - \frac{1}{r}}{1 + \frac{3}{r} + \frac{3}{r^2} + \frac{1}{r^3}} \). Notice how each term in this fraction is simplified, leading to easier limit evaluation.

Limit Evaluation

The final step is to evaluate the limit, especially when approaching infinity. After simplifying the expression, this step becomes straightforward.

• Consider the expression \( \lim_{r \to \infty} \frac{4 - \frac{1}{r}}{1 + \frac{3}{r} + \frac{3}{r^2} + \frac{1}{r^3}} \).
• As \( r \to \infty \), terms like \( \frac{1}{r} \), \( \frac{1}{r^2} \), and \( \frac{1}{r^3} \) approach zero because a very large denominator makes the fraction very small.
• This simplifies our expression to \( \frac{4 - 0}{1 + 0 + 0 + 0} = 4 \).

With this knowledge, it becomes clear why the limit evaluates to \( 4 \). The procedure highlights the importance of focusing on dominant terms, simplifying expressions, and understanding how these lead to the final limit outcome.