91Ó°ÊÓ

When dealing with sequences, identifying the dominant term is crucial for determining the behavior as the sequence approaches infinity. A dominant term is the term in the sequence that grows significantly faster than others when the variable, often denoted as \( n \), becomes very large.
For example, in the sequence \( a_{n}=\frac{6^{n}+3^{n}}{6^{n}+n^{100}} \), we have expressions in both the numerator and the denominator. To simplify and find limits, compare how different terms grow:

• In the numerator: \( 6^n \) grows faster than \( 3^n \).
• In the denominator: \( 6^n \) also dominates over \( n^{100} \).

Why is this important? Because focusing on the dominant terms helps simplify the sequence and makes it easier to understand.
This simplification step allows you to factor out these dominant terms and isolate the significant components affecting the growth rate of the sequence.

Factoring is a powerful technique in calculus that allows us to simplify expressions, especially when evaluating limits of sequences or functions. By factoring out the dominant term, we can often remove complexity and directly observe how the sequence behaves. Let’s see how this works in the given sequence:
After identifying \( 6^n \) as the dominant term:

• Factor it out from both the numerator and the denominator: \( a_{n}=\frac{6^{n}(1+\frac{3^n}{6^n})}{6^{n}(1+\frac{n^{100}}{6^n})} \).
• Cancelling \( 6^n \) simplifies the sequence to \( \frac{1+\frac{3^n}{6^n}}{1+\frac{n^{100}}{6^n}} \).

This expression is much more manageable. Factoring helps us reduce the problem to a simpler form where we can more easily analyze the limit behavior.
Remember, factoring in calculus is not just a mechanical step. It is about revealing the significant components of the expression, which helps in understanding the mathematical behavior of sequences as they grow large.

Understanding the relationship between exponents and limits is key to evaluating the behavior of sequences. Exponents indicate how rapidly terms in a sequence grow, and this understanding can be leveraged to evaluate limits efficiently. Let’s apply this to the exercise at hand:
In the simplified sequence expression, \( \frac{1+\frac{3^n}{(2\cdot3)^n}}{1+\frac{n^{100}}{(2\cdot3)^n}} \):

• The terms \( \frac{1}{2^n} \) and \( \frac{n^{100}}{2^n \cdot 3^n} \) each go to 0 as \( n \rightarrow \infty \) because these terms involve exponential growth in the denominator.

Exponential growth occurs when the base of an exponent is raised to a power. In this example, both terms in the expression decrease toward 0 because their exponential growth (2^n and (2\cdot3)^n) in the denominator outpaces the numerator.
This behavior is fundamental when calculating limits, as it helps simplify expressions to their essence — which often result in clear and simple limits, like in this case, 1. Thus, understanding exponents is crucial in deciding how individual terms in a sequence contribute to its limit.