The given image represents the following mathematical expression:

$\lim_{h \to 0} \frac{\left( \frac{9}{(a + h)^2} - \frac{9}{a^2} \right)}{h}$

This is a limit problem, and it appears to be a derivative problem involving the difference of two rational functions. To simplify and solve the limit, we can follow these steps:

Step-by-Step Solution:

1. Rewrite the expression: $\lim_{h \to 0} \frac{\left( \frac{9}{(a + h)^2} - \frac{9}{a^2} \right)}{h}$

2. Find a common denominator for the terms inside the parentheses: $\frac{9}{(a + h)^2} - \frac{9}{a^2} = 9 \left( \frac{a^2 - (a + h)^2}{(a + h)^2 a^2} \right)$ Expanding $(a + h)^2$, we get: $(a + h)^2 = a^2 + 2ah + h^2$ So, the difference becomes: $a^2 - (a^2 + 2ah + h^2) = -2ah - h^2$

3. Substitute this result back into the expression: $9 \cdot \frac{-2ah - h^2}{(a + h)^2 a^2}$

4. Simplify the entire expression: The original limit now becomes: $\lim_{h \to 0} \frac{9 \cdot \left( \frac{-2ah - h^2}{(a + h)^2 a^2} \right)}{h}$ This simplifies to: $\lim_{h \to 0} \frac{9(-2ah - h^2)}{h (a + h)^2 a^2}$ Now, divide through by $h$: $\lim_{h \to 0} \frac{9(-2a - h)}{(a + h)^2 a^2}$

5. Evaluate the limit as $h \to 0$: When $h$ approaches 0, the expression becomes: $\frac{9(-2a)}{a^4} = \frac{-18}{a^3}$

Final Answer:

$\lim_{h \to 0} \frac{\left( \frac{9}{(a + h)^2} - \frac{9}{a^2} \right)}{h} = \frac{-18}{a^3}$

Would you like more details about any of the steps?

Here are some related questions you might find helpful:

1. What is the general method for finding derivatives using limits?
2. How do you simplify complex rational expressions involving limits?
3. Why is finding a common denominator important in limit problems like this one?
4. What other types of limits can involve rational functions?
5. How does this problem relate to the concept of the derivative in calculus?

Tip: When evaluating limits that involve rational expressions, always consider factoring and simplifying the expressions to avoid indeterminate forms like $\frac{0}{0}$.