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You have uploaded an image containing the following mathematical limit:

\[
\lim_{{x \to \infty}} \frac{x^3 - 5}{x^2 - 4}
\]

### Solution:

1. To evaluate this limit as \(x\) approaches infinity, we observe that the degree of the numerator (\(x^3\)) is higher than the degree of the denominator (\(x^2\)). This typically indicates that the limit will approach infinity or some form of unbounded behavior.

2. To better understand the behavior, we divide both the numerator and the denominator by \(x^2\), the highest degree of the denominator:

\[
\frac{x^3 - 5}{x^2 - 4} = \frac{\frac{x^3}{x^2} - \frac{5}{x^2}}{\frac{x^2}{x^2} - \frac{4}{x^2}} = \frac{x - \frac{5}{x^2}}{1 - \frac{4}{x^2}}
\]

3. As \(x \to \infty\), both \(\frac{5}{x^2}\) and \(\frac{4}{x^2}\) approach 0. Thus, the expression simplifies to:

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

4. Therefore, as \(x\) approaches infinity, the limit of this expression approaches infinity:

\[
\lim_{{x \to \infty}} \frac{x^3 - 5}{x^2 - 4} = \infty
\]

### Conclusion:
The limit is \( \infty \), meaning the function grows without bound as \( x \to \infty \).

Would you like more details on this or have any further questions?

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Here are 5 related questions you might find interesting:

1. How would the limit change if the highest powers in the numerator and denominator were the same?
2. What happens to the limit if the numerator's degree is lower than the denominator's?
3. Can L'Hôpital's Rule be applied to this limit, and if so, how would the result compare?
4. What are the differences between limits at infinity and limits approaching finite values?
5. How does multiplying or dividing the entire expression by powers of \(x\) help in simplifying limits?

**Tip**: When dealing with limits at infinity, always consider the highest degree terms, as they dominate the behavior of the function.

Evaluate the limit lim (x → ∞) [(x^3 - 5)/(x^2 - 4)]

Learn how to solve the limit of (x^3 - 5)/(x^2 - 4) as x approaches infinity. This problem involves limits of rational functions, asymptotic behavior, and polynomial degree comparisons. Ideal for students in Grades 11-12 studying calculus or precalculus.

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