Problem 9 - Precalculus: Mathematical for Calculus

Question

Find the limit. $$\lim _{t \rightarrow \infty} \frac{8 t^{3}+t}{(2 t-1)\left(2 t^{2}+1\right)}$$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Identify the Dominant Terms

For the given limit \( \lim _{t \rightarrow \infty} \frac{8 t^{3}+t}{(2 t-1)(2 t^{2}+1)} \), the dominant terms are those with the highest power of \( t \) in both the numerator and the denominator. In the numerator, the dominant term is \( 8t^3 \). In the denominator, after expansion, the dominant term will be \( 4t^3 \).

Step 2: Simplify the Fraction

Divide every term in the numerator and denominator by \( t^3 \), the highest power of \( t \) found in the expression. This yields: \( \frac{8 + \frac{1}{t^2}}{4 - \frac{2}{t^2} + \frac{1}{t^3}} \).

Step 3: Evaluate the Limit as \( t \to \infty \)

As \( t \to \infty \), the terms \( \frac{1}{t^2} \) and \( \frac{1}{t^3} \) approach zero. Thus, the expression simplifies to \( \frac{8}{4} \).

Step 4: Conclude the Limit

After simplifying, we find \( \frac{8}{4} = 2 \). Therefore, the limit is \( 2 \).

Key Concepts

Dominant Terms

When working with limits in calculus, especially with polynomials, identifying dominant terms is crucial. These are the terms with the highest degree of the variable given in both the numerator and the denominator of a function. In the context of our example, the function is a rational expression, meaning it has polynomials in both the top and bottom.

Here's how you spot dominant terms:

• In the numerator, look for the term with the largest exponent of the variable. Here, it's the term \(8t^3\).
• In the denominator, consider the product of the polynomial expressions and identify the term which, after expansion, would have the highest degree. For instance, the expression \((2t-1)(2t^2+1)\) expands to include \(4t^3\), the dominant term here.

By focusing on these dominant terms, we can simplify the problem, as lower degree terms become negligible as the variable approaches infinity. This simplification is the key to evaluating limits efficiently and accurately for functions when the function variables grow very large.

Rational Functions

Rational functions are expressions of the form \( \frac{P(t)}{Q(t)} \), where \(P(t)\) and \(Q(t)\) are polynomials. They often appear in calculus problems involving limits because they can model a wide range of behaviors and tendencies of mathematical expressions.

Understanding rational functions involves:

• Recognizing their behavior as the variable approaches infinity or negative infinity. This insight often comes from the dominant terms.
• Simplifying the expression by dividing the numerator and denominator by the highest power of the variable present, as shown in our example.

This method helps transform the limit problem into a simpler form where the impact of smaller terms fades away, and you're left with a clear understanding of the overall behavior as values become very large.

Evaluating Limits at Infinity

Evaluating limits at infinity involves determining the value a function approaches as the variable grows towards infinity. This is a very common process in calculus, especially with rational functions.

In our example, once you've identified the dominant terms and simplified the function by dividing through by the highest power, evaluating becomes straightforward:

• Recognize that terms with the variable in the denominator become negligible. For instance, \(\frac{1}{t^2}\) and \(\frac{1}{t^3}\) approach zero as \(t\) approaches infinity.
• What remains are the dominant terms, \( \frac{8 + 0}{4 - 0 + 0} = \frac{8}{4} = 2 \). This shows that as \(t\) gets very large, the function approaches 2.

This approach simplifies complex function analysis and helps students understand the concept of limits in calculus by reducing the complexity while maintaining the integrity of the solution.