Question

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{6^{n}+3^{n}}{6^{n}+n^{100}}$$

Short answer

Answer: The limit of the sequence \(a_{n}\) as n approaches infinity is 1.

Step-by-step solution

Identify the dominant terms

The dominant terms are the terms with the highest exponent of n or the terms with the most significant contribution as n approaches infinity. In this case, the dominant term in the numerator is \(6^{n}\) since it grows much faster than \(3^{n}\). Similarly, the dominant term in the denominator is \(6^{n}\) as \(n^{100}\) grows more slowly in comparison. #Step 2: Divide the numerator and the denominator by the dominant term#

Divide by the dominant term

Divide both the numerator and the denominator by \(6^{n}\). $$ \frac{ \frac{6^{n} + 3^{n}}{6^{n}} }{ \frac{6^{n} + n^{100}}{6^{n}} } $$ #Step 3: Simplify the expression#

Simplify the expression

After dividing by \(6^{n}\), the expression becomes: $$ \frac{ 1 + \left(\frac{3}{6}\right)^{n} }{ 1 + \frac{n^{100}}{6^{n}} } $$ #Step 4: Find the limit#

Find the limit as n approaches infinity

As n approaches infinity, \(\left(\frac{3}{6}\right)^{n}\) approaches 0 because \((\frac{1}{2})^{n}\) goes to 0 while \(n\to\infty\). Similarly, the term \(\frac{n^{100}}{6^{n}}\) approaches 0 since the exponential function grows much faster than the polynomial function. The limit becomes: $$ \lim_{n \to \infty} \frac{ 1 + \left(\frac{3}{6}\right)^{n} }{ 1 + \frac{n^{100}}{6^{n}} } = \frac{ 1 + 0 }{ 1 + 0 } = \frac{1}{1} = 1 $$ The limit of the sequence \(a_{n}\) as n approaches infinity is 1.

Key concepts

Limits

Calculus often involves finding the limit of a sequence or function as it approaches a particular point or infinity. A limit examines the behavior of the terms in a sequence or the values a function produces when the input gets very large or very close to a specific point.

In this exercise, we are dealing with the sequence \(a_n = \frac{6^n + 3^n}{6^n + n^{100}}\). We are asked to find what happens to this sequence as \(n\) becomes very large.

• A crucial first step is determining which terms grow fastest as \(n\) increases because these terms influence the limit the most.
• "Dominant terms" play a key role, as they dictate the sequence's behavior at infinity.

By identifying and simplifying with these dominant terms, we can find that the limiting value of the sequence is 1 as \(n\) goes to infinity. Limits help us understand the long-term behavior of sequences and functions, giving insights into their stability and end behaviors.

Sequences

Sequences are ordered lists of numbers defined by an explicit formula. Each number in the sequence is called a term. The sequence can be finite or infinite depending on how we define it.

For example, the sequence given in this problem, \(a_n = \frac{6^n + 3^n}{6^n + n^{100}}\), is an infinite sequence as it progresses to infinity.

• Each term in the sequence is identified with an integer index \(n\).
• The behavior of such a sequence as \(n\) becomes extremely large is often of interest, as it can show trends and asymptotic behavior.

Analyzing sequences extensively is crucial in calculus and mathematical analysis. This is because they offer a foundation for understanding series, calculus limits, and real-valued functions. In this exercise, observing how sequence terms simplify as \(n\) gets larger provides a glimpse into limit concepts and asymptotic properties.

Asymptotic Analysis

Asymptotic analysis is a method used in calculus and computer science to describe the behavior of functions or sequences as the input or index approaches infinity.

For the given exercise, asymptotic analysis involves understanding how the dominant term \(6^n\) dominates the sequence's growth, allowing us to simplify terms like \(\left(\frac{3}{6}\right)^n\) and \(\frac{n^{100}}{6^n}\) to zero.

• This technique relies heavily on dominance; exponential terms typically overshadow polynomial terms as \(n\) becomes very large.
• By focusing on the dominant behaviors, it becomes easier to predict and understand the sequence's behavior without exact calculations for each term.

Asymptotic analysis is a powerful technique for finding limits, understanding complexity in algorithms, and simplifying complex expressions. In this scenario, the attention is on catching how each term's magnitude influences the entire expression's behavior, helping us compute limits seamlessly as \(n\) approaches infinity.