Problem 11 - Precalculus: Mathematics 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: Write the Limit Expression

We need to evaluate the limit \( \lim _{t \rightarrow \infty} \frac{8 t^{3}+t}{(2 t-1)(2t^{2}+1)} \). The first step in solving a limit problem is to properly express the mathematical limit to understand the problem’s structure.

Step 2: Identify the Dominant Terms

In the numerator, the dominant term as \( t \to \infty \) is \( 8t^3 \). In the denominator, distribute to find the dominant term: \[ (2t-1)(2t^2+1) = 4t^3 - 2t + 2t^2 - 1 \] The dominant term in the denominator is \( 4t^3 \).

Step 3: Simplify the Limit Expression

Focus on the dominant terms: \[ \lim_{t \to \infty} \frac{8t^3 + t}{4t^3} \] Simplify by dividing both numerator and denominator by \( t^3 \): \[ = \lim_{t \to \infty} \frac{8 + \frac{1}{t^2}}{4} \]

Step 4: Evaluate the Limit

As \( t \to \infty \), \( \frac{1}{t^2} \to 0 \). Therefore, the limit simplifies to: \[ \frac{8 + 0}{4} = 2 \]

Step 5: Conclusion

Conclude that after evaluating the simplification, the limit of the original expression as \( t \to \infty \) is \( 2 \).

Key Concepts

Dominant Terms in Limits

When we evaluate limits, especially those involving infinity, it's often important to identify the dominant terms. Dominant terms are the terms that increase or decrease the fastest, having the most significant impact on the behavior of a function as the variable approaches infinity. In our example, we looked at the limit of the expression \( \frac{8t^3 + t}{(2t-1)(2t^2+1)} \). To find which terms are dominant, examine both the numerator and the denominator separately.

• In the numerator, the term \(8t^3\) grows faster than \(t\), making it the dominant term.
• In the denominator, after expanding \((2t-1)(2t^2+1)\), the term \(4t^3\) is the dominant one.

Recognizing these dominant terms allows us to simplify the expression effectively when evaluating the limit.

Understanding Infinity in Limits

Infinity is a concept that often appears in calculus, especially when dealing with limits. When we say \(t \to \infty\), it means we're interested in what happens to a function's value as the variable \(t\) becomes very large. This can help show the end behavior of the function.
In the context of our limit problem, as \(t\) approaches infinity, the terms like \(t\) or constants in the expression become negligible compared to powers of \(t\) like \(t^3\). Therefore, when dealing with terms at infinity, we focus on those terms that grow the fastest (dominant terms), as they dictate the limit of the entire function.

Simplification of Limit Expressions

Simplification is a crucial step in solving limit problems. Once we've identified the dominant terms, we can simplify the limit expression to make calculations easier. In the given example, after identifying \(8t^3\) and \(4t^3\) as dominant terms in the numerator and denominator respectively, we then simplify by dividing both terms by \(t^3\). This simplification leads to:

• \(\lim_{t \to \infty} \frac{8 + \frac{1}{t^2}}{4}\)

This step eliminates all negligible terms, focusing only on terms that affect the limit, thereby making it easier and clearer to evaluate the overall limit.

Evaluating the Limit After Simplification

The final step in solving a limit problem is to evaluate the simplified expression as the variable approaches its target (in this case, infinity). After simplifying to \(\frac{8 + \frac{1}{t^2}}{4}\), we can evaluate the limit for \(t \to \infty\).

• As \(t\) grows, \(\frac{1}{t^2}\), which was once part of our expression, approaches zero.
• This leaves us with the simpler expression: \(\frac{8 + 0}{4}\).

Therefore, the limit as \(t\) approaches infinity is 2. By systematically identifying dominant terms and simplifying, we reach a straightforward answer for the limit.