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