Problem 1 - Calculus Single Variable

Question

In each of Exercises \(1-20,\) evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\frac{2 n^{2}-1+n^{3}}{5 n^{3}+n+2} $$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{5} \).

Step 1: Identify the Dominant Terms

In the given sequence \( a_n = \frac{2n^2 - 1 + n^3}{5n^3 + n + 2} \), identify the terms with the highest degree in the numerator and the denominator. These are \( n^3 \) in the numerator and \( 5n^3 \) in the denominator.

Step 2: Simplify the Expression

Divide each term in the numerator and the denominator by the highest degree term \( n^3 \) to get:\[ a_n = \frac{\frac{2n^2}{n^3} - \frac{1}{n^3} + \frac{n^3}{n^3}}{\frac{5n^3}{n^3} + \frac{n}{n^3} + \frac{2}{n^3}} = \frac{\frac{2}{n} - \frac{1}{n^3} + 1}{5 + \frac{1}{n^2} + \frac{2}{n^3}} \]

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

As \( n \to \infty \), the terms \( \frac{2}{n} \), \( \frac{1}{n^3} \), \( \frac{1}{n^2} \), and \( \frac{2}{n^3} \) all approach 0. The expression simplifies to:\[ \lim_{n \to \infty} a_n = \frac{0 + 0 + 1}{5 + 0 + 0} = \frac{1}{5} \]

Key Concepts

Dominant Terms

To evaluate the limit of a sequence, a fundamental step is identifying what's called the "dominant term." The dominant term is the term in the expression that increases the most as the sequence approaches infinity. In polynomial expressions, this is usually determined by the term with the highest degree, or exponent. For example, in the sequence \(a_n = \frac{2n^2 - 1 + n^3}{5n^3 + n + 2}\), in the numerator, the term \(n^3\) dominates because it has the highest degree of 3. Similarly, in the denominator, the leading term is \(5n^3\), which also has a degree of 3. Recognizing these terms simplifies the process of finding limits because as \(n\) becomes very large, these terms will govern the behavior of the entire expression.

Simplifying Expressions

Once the dominant terms are identified, simplifying the expression by dividing all terms by the highest degree term is the next step. Simplification reduces complex expressions into a form that is much easier to work with, especially when finding limits. In our example, every term in \(a_n = \frac{2n^2 - 1 + n^3}{5n^3 + n + 2}\) is divided by \(n^3\), the highest degree term. This gives:

• Numerator: \(\frac{2}{n} - \frac{1}{n^3} + \frac{n^3}{n^3} = \frac{2}{n} - \frac{1}{n^3} + 1\)
• Denominator: \(5 + \frac{1}{n^2} + \frac{2}{n^3}\)

In this simplified form, it is easier to see how the smaller terms decrease as \(n\) approaches infinity, aiding in the accurate evaluation of the limit.

Evaluating Limits

The process of evaluating limits involves considering the behavior of a sequence as \(n\) approaches infinity. Once an expression is simplified, evaluating the limit involves observing which terms effectively 'vanish.' As \(n\) grows infinitely large, terms like \(\frac{2}{n}\), \(\frac{1}{n^3}\), \(\frac{1}{n^2}\), and \(\frac{2}{n^3}\) shrink towards zero. Therefore, they become negligible. Using the simplified form of \(a_n = \frac{\frac{2}{n} - \frac{1}{n^3} + 1}{5 + \frac{1}{n^2} + \frac{2}{n^3}}\), as \(n\) tends to infinity, the expression becomes:

• Numerator simplifies to \(0 + 0 + 1 = 1\)
• Denominator simplifies to \(5 + 0 + 0 = 5\)

This results in the final limit \(\lim_{n \to \infty} a_n = \frac{1}{5}\). Such detailed steps ensure a clear understanding of how dominant terms shape the solution as sequences grow.