This is a limit problem that resembles the definition of the derivative of a function at a point. The given expression is:

$\lim_{h \to 0} \frac{(9 + h)^{-1} - 9^{-1}}{h}$

This is very similar to the definition of the derivative:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Here, we have $f(x) = x^{-1}$, and it seems like the point where we are calculating the derivative is $x = 9$.

Let's solve it step-by-step:

Step 1: Recognize the function

The given expression represents the derivative of the function $f(x) = x^{-1}$ evaluated at $x = 9$. The derivative of $f(x) = x^{-1}$ can be computed as:

$f'(x) = -x^{-2}$

Step 2: Compute the derivative at $x = 9$

The derivative of $f(x) = x^{-1}$ at $x = 9$ is:

$f'(9) = -9^{-2} = -\frac{1}{81}$

Step 3: Conclusion

Thus, the value of the limit is:

$\lim_{h \to 0} \frac{(9 + h)^{-1} - 9^{-1}}{h} = -\frac{1}{81}$

If you'd like further details or have questions, feel free to ask!

Here are 5 related questions:

1. What is the formal definition of a derivative using limits?
2. How would this limit change if we had a different function, like $f(x) = x^2$?
3. What happens if we substitute different values for $x$ in the original function $f(x) = x^{-1}$?
4. Can this limit be solved using L'Hopital's Rule? Why or why not?
5. How can the power rule of derivatives be applied to functions of the form $f(x) = x^n$?

Tip: Remember, recognizing the structure of the problem as the derivative of a function can significantly simplify complex-looking limit problems.