Question

Evaluate the limit of the following sequences. $$a_{n}=\frac{n^{8}+n^{7}}{n^{7}+n^{8} \ln n}$$

Short answer

Answer: The limit of the given sequence as n approaches infinity is 0.

Step-by-step solution

Simplify the expression

First, factor out \(n^{7}\) from both the numerator and the denominator: $$a_{n} = \frac{n^{7}(n + 1)}{n^{7}(1+n\ln n)}$$ Then, divide both the numerator and the denominator by \(n^{7}\): $$a_{n} = \frac{n+1}{1 + n\ln n}$$

Analyze the dominating terms

As n approaches infinity, the dominating terms of the numerator and the denominator are the terms with the highest power of n. In this case, they are: Numerator: \(n\) Denominator: \(n\ln n\)

Evaluate the limit using L'Hopital's Rule

To evaluate the limit, we can apply L'Hôpital's Rule since both the numerator and the denominator approach infinity as n approaches infinity: $$\lim_{n\to\infty} a_n = \lim_{n\to\infty} \frac{n+1}{1 + n\ln n} = \lim_{n\to\infty} \frac{(n+1)'}{(1 + n\ln n)'}$$ Now, we need to find the derivative of the numerator and the denominator with respect to n: Numerator derivative: \((n+1)' = 1\) Denominator derivative: \((1 + n\ln n)' = 0 + (\ln n + 1)\) So, the limit becomes: $$\lim_{n\to\infty} a_n = \lim_{n\to\infty} \frac{1}{\ln n + 1}$$

Finalize the limit

As n approaches infinity, the denominator (\(\ln n + 1\)) approaches infinity, while the numerator remains constant at 1. Therefore, the limit of the sequence is: $$\lim_{n\to\infty} a_n = \frac{1}{\infty} = 0$$ Thus, the limit of the given sequence is 0.

Key concepts

L'Hôpital's Rule

When evaluating limits, especially if both the numerator and the denominator approach zero or infinity, L'Hôpital's Rule becomes very useful. This rule states that for functions \( f(x) \) and \( g(x) \), if both approach 0 or \( \pm \infty \) as \( x \to c \), then: \[\lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c} \frac{f'(x)}{g'(x)}\] provided that the limit on the right exists. The rule helps simplify complex expressions by focusing on the growth rates of their derivatives. For the sequence \( a_n = \frac{n+1}{1+n\ln n} \), as \( n \) approaches infinity, both the numerator \( n+1 \) and the denominator \( 1+n\ln n \) tend towards infinity. The derivatives in this case simplify the expression to \( \lim_{n\to\infty} \frac{1}{\ln n + 1} \). This simplification makes it clear how the expression behaves as \( n \) becomes very large. The use of derivatives in L'Hôpital's Rule often turns such indeterminate forms into limits that can be directly evaluated, as we did here.

Dominating Terms

The concept of dominating terms helps identify which parts of algebraic expressions grow the fastest as variables approach infinity. In large \( n \) values, not all terms contribute equally to a function's behavior. In our exercise, for the function \( \frac{n+1}{1+n\ln n} \), as \( n \to \infty \), - In the numerator: \( n \) dominates over constant \( 1 \).- In the denominator: \( n\ln n \) dominates over constant \( 1 \).This hierarchy of terms highlights the most significant parts of the expression affecting its limit. Understanding which terms dominate allows us to simplify our computation by focusing on the highest order terms only. This simplification eventually leads to applying techniques like L'Hôpital's Rule. By prioritizing dominating terms, we manage large expressions effectively, which is particularly useful in calculus when dealing with infinite sequences or series.

Infinity in Calculus

In calculus, infinity often represents growth without bound. It denotes an unbounded limit or sequence and is crucial in evaluating the behavior of functions and sequences as variables grow. Here, we see infinity play a role when evaluating \( \frac{n+1}{1+n\ln n} \). As \( n \to \infty \), both the numerator and denominator stretch towards infinity, prompting the use of L'Hôpital's Rule. Infinity leads us to evaluate limits by observing how different terms grow relative to each other. It's through understanding these infinite behaviors that we derive actual numeric limits for sequences like the one in this example. As seen, infinity leads us to evaluate the end behavior of a sequence by teasing out which part of a sequence grows faster, defining its eventual convergence (in this case towards 0). Consequently, mastering infinity's intricacies is key to unlocking advanced calculus concepts and solving real-world problems.