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To find the limit

\[
\lim_{x \to \infty} \frac{x^3 - 3}{5x^3}
\]

we can simplify the expression by dividing both the numerator and the denominator by \(x^3\):

\[
\lim_{x \to \infty} \frac{x^3 - 3}{5x^3} = \lim_{x \to \infty} \frac{x^3/x^3 - 3/x^3}{5x^3/x^3}
\]

This simplifies to:

\[
\lim_{x \to \infty} \frac{1 - \frac{3}{x^3}}{5}
\]

As \( x \) approaches infinity, the term \( \frac{3}{x^3} \) approaches 0. So, the expression further simplifies to:

\[
\frac{1 - 0}{5} = \frac{1}{5}
\]

Thus, the limit is:

\[
\lim_{x \to \infty} \frac{x^3 - 3}{5x^3} = \frac{1}{5}
\]

Would you like more details or have any further questions?

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**Related Questions:**

1. How do you find limits at infinity for polynomial functions?
2. What is L'Hôpital's Rule, and when can it be applied?
3. How do you handle limits involving indeterminate forms?
4. Can you explain the difference between limits at infinity and limits as \( x \) approaches a finite number?
5. What is the significance of horizontal asymptotes in limits at infinity?
6. How does the degree of the numerator compare to the degree of the denominator in determining the limit at infinity?
7. What happens if the degrees of the numerator and the denominator are equal?
8. How do you interpret a limit that equals infinity?

**Tip:** Simplifying the expression by dividing both numerator and denominator by the highest power of \( x \) is often a good first step when dealing with limits at infinity.

Learn how to calculate the limit of the function (x^3 - 3) / (5x^3) as x approaches infinity with a step-by-step solution. This problem involves understanding limits at infinity and the simplification techniques used for polynomial functions. Suitable for advanced high school or college-level students.

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