https://file.aiquickdraw.com/mathbot/file/1726324563875pz1m0tcp.jpg

The limit presented in the image is:

\[
\lim_{n \to \infty} \frac{n^3 - n}{\sqrt{1 + n^6}}
\]

To solve this limit, we need to analyze the behavior of the numerator and the denominator as \( n \) approaches infinity. Notice that in the denominator, \( n^6 \) dominates, so \( \sqrt{1 + n^6} \) behaves like \( n^3 \) for large \( n \).

Therefore, the expression simplifies to:

\[
\lim_{n \to \infty} \frac{n^3 - n}{n^3} = \lim_{n \to \infty} \left(1 - \frac{1}{n^2}\right)
\]

As \( n \to \infty \), \( \frac{1}{n^2} \to 0 \), so the limit is:

\[
1 - 0 = 1
\]

It seems there might be an error with the options provided in the image as none of them match the correct answer. The correct answer should be \( 1 \), not one of the choices (a) -1, (b) (crossed out), (c) 0, or (d).

Do you have any other questions or need further explanation on this problem? Here are some additional questions you could consider:

1. How would the limit change if \( n \) approached negative infinity instead?
2. What would happen to the limit if the numerator was \( n^3 + n \) instead?
3. How does the dominant power of \( n \) in the numerator and denominator influence the limit?
4. Can you find the limit if \( n^6 \) in the denominator is replaced with \( n^4 \)?
5. What are other techniques that can be used to evaluate such limits?

Tip: When evaluating limits involving large powers of \( n \), factor out the highest power from both the numerator and the denominator to simplify the expression.

Find the limit of (n^3 - n) / sqrt(1 + n^6) as n approaches infinity.

Learn how to find the limit of the expression (n^3 - n) / sqrt(1 + n^6) as n approaches infinity. This problem explores the dominance of powers and simplifying limits of rational functions. Suitable for advanced high school or college-level students.

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