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.