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The given mathematical expression is:

\[
\lim_{{x \to 0}} \frac{x^4}{9 + x^{-1}}
\]

Let's evaluate this limit step by step.

### Step 1: Analyze the expression as \( x \to 0 \)

As \( x \to 0 \), we observe the behavior of both the numerator and denominator:

- **Numerator**: \( x^4 \) tends to 0.
- **Denominator**: The term \( x^{-1} = \frac{1}{x} \) tends to infinity (specifically, positive infinity if \( x \to 0^+ \) and negative infinity if \( x \to 0^- \)).

So the denominator becomes dominated by the \( x^{-1} \) term as \( x \to 0 \), which tends to infinity. Thus, the denominator behaves like a very large positive or negative value depending on the direction of the limit.

### Step 2: Limit conclusion

- The numerator \( x^4 \to 0 \).
- The denominator \( 9 + x^{-1} \) tends to \( \infty \) as \( x \to 0^+ \) or \( -\infty \) as \( x \to 0^- \).

Therefore, the overall limit is:

\[
\lim_{{x \to 0}} \frac{x^4}{9 + x^{-1}} = 0
\]

### Conclusion

The value of the limit is:

\[
0
\]

---

Would you like more details on the reasoning? Here are five questions that could expand your understanding:

1. What happens if we approach the limit from the negative side (\( x \to 0^- \))?
2. Why does \( x^{-1} \) dominate the denominator as \( x \to 0 \)?
3. How can L'Hopital's Rule be used to evaluate limits of indeterminate forms?
4. How would the limit change if we modified the exponent in \( x^4 \)?
5. What role does the constant 9 in the

Evaluate the limit lim (x -> 0) (x^4 / (9 + x^-1)).

In this problem, learn how to evaluate the limit of (x^4 / (9 + x^-1)) as x approaches 0. This solution covers concepts in calculus, specifically analyzing behavior as x approaches infinity or zero.

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