Question

In the following problems, find the limit of the given sequence as \(n \rightarrow \infty\). $$\frac{n^{2}+5 n^{3}}{2 n^{3}+3 \sqrt{4+n^{6}}}$$

Short answer

1

Step-by-step solution

Identify the Dominant Terms

Consider the terms with the highest powers in the numerator and the denominator. For the given sequence \(\frac{n^2+5n^3}{2n^3+3\sqrt{4+n^6}}\), the dominant terms are \(5n^3\) in the numerator and \(2n^3\) in the denominator because as \(n \rightarrow \infty\), these terms will grow significantly larger than the others.

Simplify the Expression

Divide both the numerator and the denominator by \(n^3\) to make the dominant terms more apparent: \[\frac{\frac{n^2}{n^3} + \frac{5n^3}{n^3}}{\frac{2n^3}{n^3} + \frac{3\sqrt{4+n^6}}{n^3}} = \frac{\frac{1}{n} + 5}{2 + 3\frac{\sqrt{4+n^6}}{n^3}}\].

Simplify Further

Observe that \(\frac{1}{n} \rightarrow 0\) as \(n \rightarrow \infty\). Now simplify the denominator term \(3\frac{\sqrt{4+n^6}}{n^3}\). Note that \(\sqrt{4+n^6} = n^3\sqrt{\frac{4}{n^6} + 1}\), leading to \[3\frac{\sqrt{4+n^6}}{n^3} = 3\frac{n^3 \sqrt{\frac{4}{n^6} + 1}}{n^3} = 3\sqrt{\frac{4}{n^6} + 1}\].\ As \(n \rightarrow \infty, \sqrt{\frac{4}{n^6} + 1} \rightarrow 1\). So \(3\frac{\sqrt{4+n^6}}{n^3} \rightarrow 3\).

Evaluate the Simplified Limit

Putting it all together, the expression now looks like: \[\frac{0 + 5}{2 + 3} = \frac{5}{5} = 1\].

Conclusion

Thus, the limit of the sequence as \(n \rightarrow \infty\) is 1.

Key concepts

Dominant Terms

When dealing with sequences and limits, dominant terms are those with the highest degree in polynomials or the fastest-growing terms in composite functions.
In our exercise, the sequence is given: \(\frac{n^2 + 5n^3}{2n^3 + 3\text{sqrt}{4+n^6}} \).
As we simplify to find the limit as \(n \rightarrow \infty\), we focus on the terms that grow the fastest.
This means identifying the highest power of \(n\) in both the numerator and the denominator.
Here:

• Numerator: 5n^3 (since it grows faster than n^2).
• Denominator: 2n^3 (since it grows faster than the other terms).

This allows us to zero in on these terms for simplification.

Infinite Limits

Infinite limits help us understand the behavior of sequences as \(n\) grows indefinitely.
In simpler terms, we look at what the sequence approaches when \(n\) becomes very large.
In our example, we first identified the dominant terms.
Then, we simplified the sequence to focus on how the numerator and denominator behave.
As \(n\) goes to infinity, some terms become negligible (like \(\frac{1}{n}\) approaching 0).
So, we adjust the sequence for these infinite limits, simplifying our calculations.

Numerator and Denominator Simplification

Simplifying the numerator and denominator helps in evaluating complex sequences.
Let's simplify: \(\frac{n^2 + 5n^3}{2n^3 + 3\sqrt{4+n^6}}\) by dividing both by \(n^3\):

• Numerator: \(\frac{n^2}{n^3} + \frac{5n^3}{n^3} = \frac{1}{n} + 5\).
• Denominator: \(\frac{2n^3}{n^3} + \frac{3\sqrt{4 + n^6}}{n^3}\).\

Note, \(3\sqrt{4 + n^6}/{n^3}\) simplifies to \(3\sqrt{\frac{4}{n^6} + 1}\), as \(n \rightarrow \infty\), \(3\sqrt{\frac{4}{n^6} + 1} \rightarrow 3\).
We put it all together as: \(\frac{\frac{1}{n} + 5}{2 + 3}\) which further simplifies to: \(\frac{0 + 5}{2 + 3} = 1\).

Asymptotic Behavior

Understanding asymptotic behavior is key in determining sequence limits.
It tells us how a function behaves as the input grows very large.
In our problem: \(\frac{n^2 + 5n^3}{2n^3 + 3\sqrt{4+n^6}}\), the dominant terms \(5n^3\) and \(2n^3\) show how the sequence behaves as \(n\) approaches infinity.
By focusing on these dominant terms, we simplified our understanding of the sequence.
It boils down to observing that other terms become negligible.
This process leads us to the final simplification: \(\frac{5}{5} = 1\).
Hence, the sequence's limit is 1, showcasing its asymptotic behavior.