Problem 7 - Calculus Single Variable

Question

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\frac{6 n+\cos (n)}{3 n+2-\sin \left(n^{2}\right)} $$

Step-by-Step Solution

Answer

The limit of the sequence is 2.

Step 1: Analyze the Sequence

The given sequence is \(a_n = \frac{6n + \cos(n)}{3n + 2 - \sin(n^2)}\). Notice that both the numerator and the denominator are polynomials mixed with trigonometric functions. As \(n\) approaches infinity, the polynomial terms \(6n\) and \(3n\) in the numerator and denominator will dominate the behavior of the sequence.

Step 2: Simplify the Dominant Terms

Identify the dominant terms in both the numerator and the denominator. The dominant terms are the linear terms: \(6n\) in the numerator and \(3n\) in the denominator. As \(n\) becomes very large, \(\cos(n)\) and \(-\sin(n^2)\) will oscillate between -1 and 1, becoming negligible in the context of the dominant linear terms.

Step 3: Factor Out the Dominant Terms

Factor \(n\) out of the numerator and the denominator.\[ a_n = \frac{n(6 + \frac{\cos(n)}{n})}{n(3 + \frac{2}{n} - \frac{\sin(n^2)}{n})} \]By canceling the \(n\) terms from the numerator and denominator, we simplify the expression.

Step 4: Take the Limit

Take the limit as \(n\) approaches infinity. As \(n \to \infty\), the terms \(\frac{\cos(n)}{n}\), \(\frac{2}{n}\), and \(\frac{\sin(n^2)}{n}\) all approach 0.\[ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{6 + \frac{\cos(n)}{n}}{3 + \frac{2}{n} - \frac{\sin(n^2)}{n}} = \frac{6}{3} = 2 \]

Step 5: Conclusion

The limit of the sequence \(\{a_n\}\) as \(n\) approaches infinity is 2.

Key Concepts

Sequence Analysis

When analyzing sequences, we are often looking at how a series of numbers behaves as it progresses towards infinity. Understanding sequences involves identifying patterns or trends in the numbers. In calculus, sequence analysis helps us to predict the value a sequence will approach, called the limit.

• A sequence consists of ordered elements based on some rule or formula.
• Analyzing a sequence involves finding the dominant elements that influence its behavior as it approaches very large numbers.
• For sequences like the one given, inspecting the structure of the terms helps in determining which will grow quickest and influence the sequence's behavior at infinity.

By focusing on the dominant terms, we can often simplify complex sequences and easily compute their limits.

Dominant Terms

In the context of limits, dominant terms are those which grow most significantly as the variable approaches infinity. Usually, these are the terms with the highest degree or fastest rate of increase.Trigonometric functions like \(\cos(n)\) and \(-\sin(n^2)\) oscillate within a fixed range, and their contributions become negligible compared to polynomial terms as \(n \to\ infinity\).

• The sequence in the exercise includes polynomial and trigonometric functions.
• By identifying \(6n\) and \(3n\) as dominant, you focus on the linear terms.
• Oscillating terms like \(cos(n)\) and \(-\sin(n^2)\) do not impact the limit as \(n\) grows very large.

Thus, the behavior of the sequence at infinity is driven by the dominant linear terms, simplifying our calculations.

Trigonometric Functions

Trigonometric functions such as sine and cosine often appear in mathematical expressions and have unique behaviors due to their oscillation. They oscillate within a set interval, \(-1\) to \(1\), over a cycle, impacting the limit of sequences in limited ways.

• Sine and cosine functions oscillate and do not grow towards infinity.
• This inherent behavior allows them to be less influential in the determination of dominant terms for limits.
• When paired with polynomial functions, trigonometric functions tend to fade in importance as \(n\) grows larger.

In the limit of the given sequence, the sine and cosine terms quickly become negligible, allowing for easier simplification when compared to polynomial components.

Polynomial Behavior

Polynomials often drive the behavior of sequences and series, especially in real-world applications. Their highest-degree term usually determines their growth rate. In limits, keeping an eye on the polynomial degree helps predict the sequence's direction as the variables become exceedingly large.

• Polynomials consist of terms with variables raised to integer powers.
• When approaching infinity, the term with the highest degree is most significant.
• In our sequence's numerator and denominator, the linear terms \(6n\) and \(3n\) are crucial in understanding the sequence's behavior at infinity.

By focusing on the dominant polynomial terms, evaluating the sequence becomes straightforward, simplifying complex expressions into manageable calculations.