91Ó°ÊÓ

When evaluating limits, identifying dominant terms in a sequence is crucial. Dominant terms are the parts of an expression that grow the fastest as the variable approaches infinity. In our sequence \( a_n = \frac{n 2^n + 3^n + 1}{n 2^n + 3^n + 2} \),the dominant term is found by comparing how quickly each term grows as \( n \) becomes very large. Here,

• The term \( 3^n \) grows very quickly as \( n \) increases since raising a larger base number to increasing powers grows rapidly.
• The term \( n 2^n \), although it includes a factor that also grows, increases more slowly because multiplying \( 2^n \) by \( n \) cannot outpace the exponential growth of \( 3^n \).
• Constants like 1 and 2 become negligible compared to exponential terms.

Thus, recognizing \( 3^n \) as the dominant term simplifies the process of evaluating the limit.

Limit evaluation involves simplifying an expression to find its behavior as a variable approaches a certain point, often infinity. In calculus, this process can reveal stability or convergence in sequences or functions. In our exercise, we must simplify the original sequence to evaluate the limit:\[a_n = \frac{n 2^n + 3^n + 1}{n 2^n + 3^n + 2}\]After identifying \( 3^n \) as the dominant term in both the numerator and the denominator, we divide every component by this term:\[\frac{\frac{n 2^n}{3^n} + 1 + \frac{1}{3^n}}{\frac{n 2^n}{3^n} + 1 + \frac{2}{3^n}}\]In doing so, we focus on how each part behaves. As \( n \) approaches infinity,

• Expressions like \( \frac{n 2^n}{3^n} = n \left( \frac{2}{3} \right)^n \) diminish rapidly to zero because \( \left( \frac{2}{3} \right)^n \) converges to zero faster than \( n \) diverges.
• Other terms involving \( \frac{1}{3^n} \) and \( \frac{2}{3^n} \) effectively shrink to zero, leaving 1 as the remaining significant term.

This approach simplifies the sequence to \( \frac{1}{1} \), allowing us to conclude that \( \lim_{n \to \infty} a_n = 1 \).

The process of simplifying sequences involves reducing complex expressions into simpler forms to facilitate understanding, especially when evaluating limits. In this exercise, we've seen

• How breaking down terms by their growth rates prepares expressions for division by dominant terms, which significantly simplifies the limit evaluation.
• After simplifying via division by the dominant term, we simplify further by acknowledging terms that disappear, like \( \left( \frac{2}{3} \right)^n \), because they reduce to zero.
• This transforms the sequence into its simplest form where unstable or negligible parts vanish as \( n \to \infty \).

Simplifying the sequence to its core components, such as 1 here, one can easily evaluate the limit. Mastering this technique allows students to solve similar problems efficiently, focusing on the essential growth behaviors of sequence terms.