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Chapter 9: Problem 11

Find the limit of the following sequences or determine that the limit does not exist. $$\left\\{\frac{3 n^{3}-1}{2 n^{3}+1}\right\\}$$

Short Answer

Expert verified
Answer: The limit of the given sequence as n approaches infinity is \(\frac{3}{2}\).

Step by step solution

01

Identify the highest power of n in the function

In the given sequence, the highest power of n is \(n^3\) in both the numerator (\(3n^3 - 1\)) and the denominator (\(2n^3 + 1\)).
02

Divide by the highest power of n

Divide both the numerator and the denominator by the highest power of n, \(n^3\) , to simplify the function: $$\frac{3n^3 - 1}{2n^3 + 1} \cdot \frac{\frac{1}{n^3}}{\frac{1}{n^3}} = \frac{3 - \frac{1}{n^3}}{2 + \frac{1}{n^3}}$$
03

Evaluate the limit as n approaches infinity

As n approaches infinity, the terms \(\frac{1}{n^3}\) in the numerator and denominator become negligible compared to the constant terms. Therefore, we can rewrite the function as the limit as n approaches infinity: $$\lim_{n\to\infty} \frac{3 - \frac{1}{n^3}}{2 + \frac{1}{n^3}} = \frac{3}{2}$$
04

State the result

The limit of the given sequence as n approaches infinity is \(\frac{3}{2}\).

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