\lim_{{x \to \infty}} \frac{x^2}{5 - x^3}

To solve the limit

\[
\lim_{{x \to \infty}} \frac{x^2}{5 - x^3},
\]

we analyze the behavior of the function as \( x \to \infty \).

### Step 1: Identify the leading terms
As \( x \to \infty \), the highest degree term in both the numerator and denominator will dominate. In the numerator, the highest degree term is \( x^2 \), and in the denominator, it is \( -x^3 \).

Thus, the function behaves like:

\[
\frac{x^2}{-x^3} = \frac{1}{-x}.
\]

### Step 2: Take the limit
Now, evaluate the limit of \( \frac{1}{-x} \) as \( x \to \infty \):

\[
\lim_{{x \to \infty}} \frac{1}{-x} = 0.
\]

### Conclusion:
Therefore, the limit is

\[
\boxed{0}.
\]

Would you like more details, or do you have any questions about this solution?

Here are 5 related questions for further exploration:
1. What happens if the degree of the numerator and denominator are equal?
2. How do you evaluate limits involving indeterminate forms like \( \frac{\infty}{\infty} \)?
3. How would you find the limit if the exponent of \( x \) in the denominator were different?
4. What is the behavior of the function \( \frac{x^2}{5 - x^3} \) as \( x \to -\infty \)?
5. Can L'Hopital's Rule be applied to this limit, and how?

**Tip:** When solving limits involving large \( x \), always look for the highest degree terms in both the numerator and denominator to simplify the expression!

Discover how to solve the limit \lim_{{x \to \infty}} \frac{x^2}{5 - x^3} with a detailed step-by-step explanation. Learn about asymptotic behavior and the dominance of higher-degree terms in rational functions. Ideal for high school and early college students studying calculus.

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