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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.

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

Learn how to solve the limit lim_{h -> 0} ((9 + h)^-1 - 9^-1) / h using the definition of the derivative. This calculus problem involves key concepts such as limits and derivatives, showing step-by-step how to arrive at the solution of -1/81. Suitable for Grades 11-12 or college students studying calculus.

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